Reasoning in Geometry § 1.1 Patterns and Inductive Reasoning

Reasoning in Geometry § 1.1 Patterns and Inductive Reasoning
§ 1.2 Points, Lines, and Planes § 1.3 Postulates § 1.4 Conditional Statements and Their Converses § 1.5 Tools of the Trade § 1.6 A Plan for Problem Solving

2) Solve the equation. Check your answer.
5 Minute-Check 1) Both answers can be calculated Which one is right? What makes it right? What makes the other one incorrect? 2) Solve the equation. Check your answer. 3) If a dart is thrown at the circle to the right, what is the probability that it will land in a yellow sector? The odds?

6. Find the next three terms of the sequence. 6, 12, 24, . . .
5 Minute-Check Find the value or values of the variable that makes each equation true. 1. g = 21 2. x = 5 3. y = 4 or y = - 4 4. z = - 2 5. 6 6. Find the next three terms of the sequence. 6, 12, 24, . . . 48, 96, 192

Patterns and Inductive Reasoning
What You'll Learn You will learn to identify patterns and use inductive reasoning. If you were to see dark, towering clouds approaching, you might want to take cover. Your past experience tells you that a thunderstorm is likely to happen. When you make a conclusion based on a pattern of examples or past events, you are using inductive reasoning. 4

You can use inductive reasoning to find the next terms in a sequence. Find the next three terms of the sequence: 3, 6, , 24, , , X 2 X 2 X 2 X 2 X 2 Find the next three terms of the sequence: 7, 8, , 16, 23, 32 + 1 + 3 + 5 + 7 + 9 5

Draw the next figure in the pattern. 6

A _________ is a conclusion that you reach based on inductive reasoning. conjecture In the following activity, you will make a conjecture about rectangles. 1) Draw several rectangles on your grid paper. 2) Draw the diagonals by connecting each corner with its opposite corner. Then measure the diagonals of each rectangle. 3) Record your data in a table Diagonal 1 Diagonal 2 Rectangle 1 7.5 inches d1 = 7.5 in. d2 = 7.5 in. Make a conjecture about the diagonals of a rectangle 7

A conjecture is an educated guess. Sometimes it may be true, and other times it may be false. How do you know whether a conjecture is true or false? Try different examples to test the conjecture. If you find one example that does not follow the conjecture, then the conjecture is false. Such a false example is called a _____________. counterexample Conjecture: The sum of two numbers is always greater than either number. Is the conjecture TRUE or FALSE ? Counterexample: = - 2 - 2 is not greater than 3. 8

End of Lesson 9

Find the next three terms of each sequence.
5 Minute-Check Find the next three terms of each sequence. 1. 59, 63, 67 2. 15.5, , 3. Draw the next figure in the pattern shown below. 4. Find a counterexample for this statement: “The sum of two numbers is always greater than either addend.” = and 2 < 4 5) If a dart is thrown at the circle to the right, what is the probability that it will land in a shaded sector? The odds?

Points, Lines, and Planes
What You'll Learn You will learn to identify and draw models of points, lines, and planes, and determine their characteristics. Geometry is the study of points, lines, and planes and their relationships. Everything we see contains elements of geometry. Georges Seurat, Sunday Afternoon on the Island of LeGrande Jatte, Even the painting to the right is made entirely of small, carefully placed dots of color. 11

A ____ is the basic unit of geometry. point A POINT: A point has no ____. size Points are named using capital letters. B The points at the right are named point A and point B. 12

A ____is a series of points that extends without end in two directions. line LINE: A line is made up of an ______ _______ of points. infinite number The ______ show that the line extends without end in both directions. arrows A line can be named with a single lowercase script letter or by two points on the line. The line below is named line AB, line BA, or line l. The symbol for line AB is A B l 13

1) Name two points on line m. R T m S Possible answers: point R and point S point R and point T point S and point T 2) Give three names for the line. Possible answers: or line m NOTE: Any two points on the line or the script letter can be used to name it. 14

Three points may lie on the same line. These points are _______ . collinear Points that DO NOT lie on the same line are __________ . noncollinear R T S U V 1) Name three points that are collinear. Possible answers: points R, S, and point T points U, S, and point V 15

Three points may lie on the same line. These points are _______ . collinear Points that DO NOT lie on the same line are __________ . noncollinear R T S U V 1) Name three points that are noncollinear. Possible answers: points R, S, and point V points R, T, and point U points R, S, and point U points R, T, and point V points R, V, and point U points S, T, and point V 16

Rays and line segments are parts of lines. A ___ has a definite starting point and extends without end in one direction. ray A B RAY: The starting point of a ray is called the ________. endpoint A ray is named using the endpoint first, then another point on the ray. The ray above is named ray AB. The symbol for ray AB is 17

Rays and line segments are parts of lines. A ___________ has a definite beginning and end. line segment LINE SEGMENT: A line segment is part of a line containing two endpoints and all points between them. A B A line segment is named using its endpoints. The line segment above is named segment AB or segment BA. The symbol for segment AB is 18

1) Name two segments. Possible Answers: D A B C U 2) Name a ray. Possible Answers: 19

A _____ is a flat surface that extends without end in all directions. plane Points that lie in the same plane are ________. coplanar Points that do not lie in the same plane are ___________. noncoplanar PLANE: For any three noncollinear points, there is only one plane that contains all three points. A M A plane can be named with a single uppercase script letter or by three noncollinear points. B C The plane at the right is named plane ABC or plane M. 20

Place points A, B, C, D, & E on a piece of paper as shown.
Hands On Place points A, B, C, D, & E on a piece of paper as shown. B D Fold the paper so that point A is on the crease. A Open the paper slightly The two sections of the paper represent different planes. C E Answers (may be others) 1) Name three points that are coplanar ______________________ A, B, & C 2) Name three points that are noncoplanar ______________________ D, A, & B 3) Name a point that is in both planes ______________________ A 21

End of Lesson 22

r 1) Name three points on line r D E C F
5-Minute Check 1) Name three points on line r D F r E C D, E, F 2) Give three other names for line r 3) Name two segments that have point F as an endpoint. 4) Name three different rays. 5) Are points C, E, and F collinear or noncollinear? noncollinear

Postulates What You'll Learn You will learn to identify and use basic postulates about points, lines, and planes. 24

Geometry is built on statements called _________. postulates
Postulates are statements in geometry that are accepted to be true. Postulate 1-1: Two points determine a unique ___. Q P line There is only one line that contains Points P and Q T l m Postulate 1-2: If two distinct lines intersect, then their intersection is a ____. point Lines l and m intersect at point T Postulate 1-3: Three noncollinear points determine a unique _____. A B C plane There is only one plane that contains points A, B, and C. 25

Points A, B, and C are noncollinear. A
Postulates Points A, B, and C are noncollinear. A 1) Name all of the different lines that can be drawn through these points. C B 2) Name the intersection of Point C 26

1) Name all of the planes that are represented in the figure.
Postulates 1) Name all of the planes that are represented in the figure. There is only one plane that contains three noncollinear points. B A D C plane ABC (side) plane ACD (side) plane ABD (back side) plane BCD (bottom) 27

Postulates Postulate 1-4: If two distinct planes intersect, then their intersection is a ___. line Plane M and plane N intersect in line DE. M N D E 28

Name the intersection of plane CDG and plane BCD.
Postulates Name the intersection of plane CDG and plane BCD. Name two planes that intersect in planes ADF and CDF H E F G C B A D 29

Postulates End of Lesson 30

5-Minute Check 1) At which point or points do three planes intersect? At each of the points A, B, C, and D. 2) Name the intersection of plane ABC and plane ACD. 3) Are there two planes in the figure that do not intersect? No 4) Name two planes that intersect in Planes ABD and BCD. B A D C 5) How many points do and have in common? One, (Point B)

Conditional Statements and Their Converses
What You'll Learn You will learn to write statements in if-then form and write the converse of the statements. In mathematics, you will come across many _______________. if-then statements For Example: If a number is even, then it is divisible by two. If – then statements join two statements based on a condition: A number is divisible by two only if the number is even. Therefore, if – then statements are also called __________ __________ . conditional statements 32

Conditional statements have two parts. The part following if is the _________ . hypothesis The part following then is the _________ . conclusion If a number is even, then the number is divisible by two. a number is even the number is divisible by two. Hypothesis: Conclusion: 33

Conditional Statements and Their Converses Conditional Statement
How do you determine whether a conditional statement is true or false? Conditional Statement True or False Why? If it is the 4th of July (in the U.S.), then it is a holiday. True The statement is true because the conclusion follows from the hypothesis. If an animal lives in the water, then it is a fish. False You can show that the statement is false by giving one counterexample. Whales live in water, but whales are mammals, not fish. 34

There are different ways to express a conditional statement. The following statements all have the same meaning. If you are a member of Congress, then you are a U.S. citizen. All members of Congress are U.S. citizens. You are a U.S. citizen if you are a member of Congress. You write two other forms of this statement: “If two lines are parallel, then they never intersect.” Possible answers: All parallel lines never intersect. Lines never intersect if they are parallel. 35

The ________ of a conditional statement is formed by exchanging the hypothesis and the conclusion. converse Conditional: If a figure is a triangle, then it has three angles. a figure is a triangle it has three angles Converse: If _______________, then ________________. NOTE: You often have to change the wording slightly so that the converse reads smoothly. Converse: If the figure has three angles, then it is a triangle. 36

Write the converse of the following statements. State whether the converse is TRUE or FALSE. If FALSE, give a counterexample: “If you are at least 16 years old, then you can get a driver’s license.” If ________________________, then _______________________. you can get a driver’s license you are at least 16 years old TRUE! “If today is Saturday, then there is no school. FALSE! If _______________, then ______________. there is no school today is Saturday We don’t have school on New Years day which may fall on a Monday. 37

End of Lesson 38

If the power goes out, we will light candles.
5-Minute Check If the power goes out, we will light candles. 1) Identify the hypothesis and conclusion of the statement. Hypothesis: the power goes out Conclusion: we will light candles 2) Write two other forms of the statement. 1) We will light candles if the power goes out. 2) Whenever the power goes out, we will light candles 3) Write the converse of the statement. If we light candles, then the power has gone out. 4) Is the converse you wrote for # 3 (above) true? NO! You could light candles for another reason, such as a birthday party.

What You'll Learn You will learn to use geometry tools.
Tools of the Trade What You'll Learn You will learn to use geometry tools. 40

As you study geometry, you will use some of the basic tools.
Tools of the Trade As you study geometry, you will use some of the basic tools. A __________ is an object used to draw a straight line. straightedge A credit card, a piece of cardboard, or a ruler can serve as a straightedge. Determine whether the sides of the triangle are straight. Place a straightedge along each side of the triangle. 41

A _______ is another useful tool. compass
Tools of the Trade A _______ is another useful tool. compass A common use for a compass is drawing arcs and circles. (an arc is part of a circle) 42

Use a compass to determine which segment is longer
Tools of the Trade Use a compass to determine which segment is longer 1) Place the point of the compass on A and adjust the compass so that the pencil is on C. 2) Without changing the setting of the compass, place the point of the compass on B. The pencil point does not reach point D. Therefore, is longer. D C B A In geometry, you will draw figures using only a compass and a straightedge. These drawings are called ___________ . constructions 43

Use a compass and straightedge to construct a six-sided figure.
Tools of the Trade Use a compass and straightedge to construct a six-sided figure. 3) Move the compass point to the arc and draw another arc along the circle Continue doing this until there are six arcs. 4) Use a straightedge to connect the points in order. 2) Using the same compass setting, put the point on the circle and draw a small arc on the circle. 1) Use the compass draw a circle. 44

Constructing the Midpoint
You will learn to construct the midpoint of a line segment using only a straightedge and compass. 1) On your patty paper, draw two points. 2) Construct a line segment between the points 3) Fold the paper, and place one point on top of the other. This should produce a crease (fold mark) between the points. 4) Place the compass on one of the points and open it to over half way to the other point. 5) Repeat step 4 using the second point. 6) Connect the intersection of the two circles. 45

Use a compass and straightedge to construct a six-sided figure.
Tools of the Trade Use a compass and straightedge to construct a six-sided figure. 3) Move the compass point to the arc and draw another arc along the circle Continue doing this until there are six arcs. 4) Use a straightedge to connect the points in order. 2) Using the same compass setting, put the point on the circle and draw a small arc on the circle. 1) Use the compass draw a circle. 46

Tools of the Trade End of Lesson

A Plan for Problem Solving
What You'll Learn You will learn to solve problems that involve the perimeters and areas of rectangles and parallelograms. Perimeter is the _____________________. distance around an object Perimeter is similar to ____________. a line segment Area is the _______________________________________________. number of square units needed to cover an object’s surface Area is similar to ______. a plane 48

In this section you will learn to solve problems that involve the perimeters and areas of rectangles and parallelograms. Perimeter is the ____________________. distance around a figure The perimeter is the ____ of the lengths of the sides of the figure. sum The perimeter of the room shown here is: 15 ft + 18 ft + 6 ft + 6 ft + 9 ft + 12 ft = 66 ft 49

Some figures have special characteristics. For example, the opposite sides of a rectangle have the same length. This allows us to use a formula to find the perimeter of a rectangle. (A formula is an equation that shows how certain quantities are related.) (of a rectangle) 50

Find the perimeter of a rectangle with a length of 17 ft and a width of 8 ft. 17 ft 8 ft (of a rectangle) = 2(17 ft) + 2(8 ft) = 2(17 ft + 8 ft) = 34 ft + 16 ft = 2(25 ft) = 50 ft = 50 ft 51

Another important measure is area. The area of a figure is ____________________________________________. the number of square units needed to cover its surface The area of the rectangle below can be found by dividing it into 18 unit squares. 3 6 The area of a rectangle can also be found by multiplying the length and the width. 52

The area “A” of a rectangle is the product of the length l and the width w. l w Find the area of the rectangle 14 in. 10 in. The area of the rectangle is 140 square inches. NOTE: units indicate area is being calculated 53

Plan for Problem Solving
Because the opposite sides of a parallelogram have the same length, the area of a parallelogram is closely related to the area of a ________. rectangle height base The area of a parallelogram is found by multiplying the ____ and the ______. base height Base – the bottom of a geometric figure. Height – measured from top to bottom, perpendicular to the base. 54

Find the area of the parallelogram: 55

§1.6 A Plan for Problem Solving
End of Lesson 56

§1.5 Tools of the Trade

Segment Measure and Coordinate Graphing
§ 2.1 Real Numbers and Number Lines § 2.2 Segments and Properties of Real Numbers § 2.3 Congruent Segments § 2.4 The Coordinate Plane § 2.5 Midpoints

1. Find the next three terms of the sequence 12, 17, 23, 30, … ,
5 Minute-Check 1. Find the next three terms of the sequence , 17, 23, 30, … , 38, 47, 57 2. Name the intersection of planes ABC and CDE in the figure. A B D E F C 3. How does a ray differ from a line? A ray extends in only one direction and has an endpoint. A line extends in two directions. 4. Find the perimeter and area of a rectangle with length of 10 centimeters and width of 4 centimeters.

Real Numbers and Number Lines
What You'll Learn You will learn to find the distance between two points on a number line. Vocabulary 1) Whole Numbers 2) Natural Numbers 3) Integers 4) Rational Numbers 5) Terminating Decimals 6) Nonterminating Decimals 7) Irrational Numbers 8) Real Numbers 9) Coordinate 10 Origin 11) Measure 12) Absolute Value 60

Numbers that share common properties can be classified or grouped into sets. Different sets of numbers can be shown on number lines. This figure shows the set of _____________ . whole numbers 2 1 4 3 6 5 7 9 8 10 The whole numbers include 0 and the natural, or counting numbers. The arrow to the right indicate that the whole numbers continue _________. indefinitely 61

This figure shows the set of _______ . integers 2 1 4 3 -5 5 -4 -2 -3 -1 negative integers positive integers The integers include zero, the positive integers, and the negative integers. The arrows indicate that the numbers go on forever in both directions. 62

A number line can also show ______________. rational numbers 2 1 -1 -2 A rational number is any number that can be written as a _______, where a and b are integers and b cannot equal ____. fraction zero The number line above shows some of the rational numbers between -2 and 2. In fact, there are _______ many rational numbers between any two integers. infinitely 63

Rational numbers can also be represented by ________. decimals Decimals may be __________ or _____________. terminating nonterminating 0.375 0.49 terminating decimals. nonterminating decimals. The three periods following the digits in the nonterminating decimals indicate that there are infinitely many digits in the decimal. 64

Some nonterminating decimals have a repeating pattern. repeats the digits 1 and 7 to the right of the decimal point. A bar over the repeating digits is used to indicate a repeating decimal. Each rational number can be expressed as a terminating decimal or a nonterminating decimal with a repeating pattern. 65

Decimals that are nonterminating and do not repeat are called _______________. irrational numbers and appear to be irrational numbers 66

____________ include both rational and irrational numbers. Real numbers 2 1 -1 -2 The number line above shows some real numbers between -2 and 2. Postulate 2-1 Number Line Postulate Each real number corresponds to exactly one point on a number line. Each point on a number line corresponds to exactly one real number 67

The number that corresponds to a point on a number line is called the _________ of the point. coordinate On the number line below, __ is the coordinate of point A. 10 The coordinate of point B is __ -4 Point C has coordinate 0 and is called the _____. origin x 11 -5 -3 -1 1 3 5 7 9 -6 -4 4 8 2 10 6 -2 B C A 68

The distance between two points A and B on a number line is found by using the Distance and Ruler Postulates. Postulate 2-2 Distance Postulate For any two points on a line and a given unit of measure, there is a unique positive real number called the measure of the distance between the points. A B measure Postulate 2-3 Ruler Postulate Points on a line are paired with real numbers, and the measure of the distance between two points is the positive difference of the corresponding numbers. B A a b measure = a – b 69

The measure of the distance between B and A is the positive difference 10 – 2, or 8. x 11 -5 -3 -1 1 3 5 7 9 -6 -4 4 8 2 10 6 -2 B A 1 2 4 3 5 7 6 8 9 10 11 Another way to calculate the measure of the distance is by using ____________. absolute value 70

Use the number line below to find the following measures. A B C F x -2 2 -1 -3 1 1 2 3 71

Use the number line below to find the following measures. A D E F x -2 2 -1 -3 1 1 2 3 72

Traveling on I-70, the Manhattan exit is at mile marker 313. The Hays exit is mile marker What is the distance between these two towns? 73

End of Lesson 74

6. Find the next three terms of the sequence. 6, 12, 24, . . .
5 Minute-Check Find the value or values of the variable that makes each equation true. 1. g = 21 2. x = 5 3. y = 4 or y = -- 4 4. z = -- 2 5. 6 6. Find the next three terms of the sequence. 6, 12, 24, . . . 48, 96, 192 75

Segments and Properties of Real Numbers
What You'll Learn You will learn to apply the properties of real numbers to the measure of segments. Vocabulary 1) Betweenness 2) Equation 3) Measurement 4) Unit of Measure 5) Precision

Given three collinear points on a line, one point is always _______ the other two points. between Definition of Betweenness Point R is between points P and Q if and only if R, P, and Q are collinear and _______________. PR + RQ = PQ P Q R PQ PR RQ NOTE: If and only if (iff) means that both the statement and its converse are true. Statements that include this phrase are called biconditionals.

Segment measures are real numbers. Let’s review some of the properties of real numbers relating to EQUALITY. Properties of Equality for Real Numbers. Reflexive Property For any number a, a = a Symmetric Property For any numbers a and b, if a = b, then b = a Transitive Property For any numbers a, b, and c, if a = b and b = c then a = c

then a may be replaced by b in any equation.
Segments and Properties of Real Numbers Segment measures are real numbers. Let’s review some of the properties of real numbers relating to EQUALITY. Properties of Equality for Real Numbers. Addition and Subtraction Properties For any numbers a, b, and c, if a = b, then a + c = b + c and a – c = b – c Multiplication and Division Properties For any numbers a, b, and c, if a = b, then a * c = b * c and a ÷ c = b ÷ c Substitution Properties For any numbers a and b, if a = b, then a may be replaced by b in any equation.

If QS = 29 and QT = 52, find ST. P Q S T QS + ST = QT QS + ST – QS = QT – QS ST = QT – QS ST = 52 – 29 = 23

If PR = 27 and PT = 73, find RT. P Q R S T PR + RT = PT PR + RT – PR = PT – PR RT = PT – PR RT = 73 – 27 = 46

End of Lesson

Refer to the figure below: Suppose AC = 49 and AB = 14.
5 Minute-Check 1. Points X, Y, and Z are collinear If XY = 32, XZ = 49, and YZ = 81, determine which point is between the other two. Y Z X Refer to the figure below: Suppose AC = 49 and AB = 14. A C B 2. Find BC. BC = AC - AB BC = BC = 35 3. Suppose D is 5 units to the right of C. What is AD? AD = AC = 54

Congruent Segments What You'll Learn You will learn to identify congruent segments and find the midpoints of segments. In geometry, two segments with the same length are called ________ _________ congruent segments Definition of Congruent Segments Two segments are congruent if and only if ________________________ they have the same length

Congruent Segments B A C P Q R S

Use the number line to determine if the statement is True or False.
Congruent Segments Use the number line to determine if the statement is True or False. Explain you reasoning. x 11 -5 -3 -1 1 3 5 7 9 -6 -4 4 8 2 10 6 -2 R S T Y Because RS = 4 and TY = 5,

These statements are called ________. theorems
Congruent Segments Since congruence is related to the equality of segment measures, there are properties of congruence that are similar to the corresponding properties of equality. These statements are called ________. theorems Theorems are statements that can be justified by using logical reasoning. 2 – 1 Congruence of segments is reflexive. 2 – 2 Congruence of segments is symmetric. 2 – 3 Congruence of segments is transitive.

There is a unique point on every segment called the _______. midpoint
Congruent Segments There is a unique point on every segment called the _______. midpoint On the number line below, M is the midpoint of What do you notice about SM and MT? x 11 -5 -3 -1 1 3 5 7 9 -6 -4 4 8 2 10 6 -2 S M T SM = MT

Definition of Midpoint
Congruent Segments Definition of Midpoint A point M is the midpoint of a segment if and only if M is between S and T and SM = MT M S T SM = MT The midpoint of a segment separates the segment into two segments of _____ _____. equal length So, by the definition of congruent segments, the two segments are _________. congruent

In the figure, B is the midpoint of . Find the value of x.
Congruent Segments In the figure, B is the midpoint of Find the value of x. 5x - 6 2x C B A Check! Since B is the midpoint: AB = BC AB = 5x – 6 = 5(2) – 6 = 10 – 6 = 4 Write the equation involving x: 5x – 6 = 2x Solve for x: 5x – 2x – 6 = 2x – 2x 3x – = 0 + 6 3x = 6 x = 2 BC = 2x = 2(2) = 4

To bisect something means to separate it into ___ congruent parts. two
Congruent Segments To bisect something means to separate it into ___ congruent parts. two The ________ of a segment bisects the segment because it separates the segment into two congruent segments. midpoint A point, line, ray, or plane can also bisect a segment. E G D B A C

Congruent Segments End of Lesson

The lengths are the same, both are 4.
5 Minute-Check 1. The lengths are the same, both are 4. 2. d + 4 = 3d = 2d 2 = d d + 4 3d S R Q 3. True; segment congruence is symmetric. 4. False; Points A, B, and C may not be collinear. 5. If a box has 5 red marbles, 5 blue marbles, and 5 green marbles, what is the probability of selecting either a blue or green marble?

The Coordinate Plane What You'll Learn You will learn to name and graph ordered pairs on a coordinate plane. In coordinate geometry, grid paper is used to locate points. The plane of the grid is called the coordinate plane. y x 5 -4 -2 1 3 -5 -1 4 -3 2

The horizontal number line is called the ______. x-axis
The Coordinate Plane The horizontal number line is called the ______. y x 5 -4 -2 1 3 -5 -1 4 -3 2 x-axis Quadrant II (–, +) Quadrant I (+, +) The vertical number line is called the ______. y-axis O The point of intersection of the two axes is called the _____. Quadrant III (–, –) Quadrant IV (+, –) origin The two axes separate the plane into four regions called _________. quadrants

Completeness Property for Points in the Plane
The Coordinate Plane An ordered pair of real numbers, called coordinates of a point, locates a point in the coordinate plane. Each ordered pair corresponds to EXACTLY ________ in the coordinate plane. one point The point in the coordinate plane is called the graph of the ordered pair. Locating a point on the coordinate plane is called _______ the ordered pair. graphing Postulate 2 – 4 Completeness Property for Points in the Plane Each point in a coordinate plane corresponds to exactly one __________________________. Each ordered pair of real numbers corresponds to exactly one __________________________. ordered pair of real numbers point in the coordinate plane

Graphing an ordered pair, (point): (x, y)
The Coordinate Plane Graphing an ordered pair, (point): (x, y) Graph point A at (4, 3) The first number, 4, is called the ___________. y x 5 -4 -2 1 3 -5 -1 4 -3 2 x-coordinate (4, 3) It tells the number of units the point lies to the __________ of the origin. left or right (0, 0) The second number, 3, is called the ___________. y-coordinate It tells the number of units the point lies _____________ the origin. above or below What is the coordinate of the origin?

Graphing an ordered pair, (point): (x, y)
The Coordinate Plane Graphing an ordered pair, (point): (x, y) Graph point B at (2, –3) The first number, 2, is called the ___________. y x 5 -4 -2 1 3 -5 -1 4 -3 2 x-coordinate It tells the number of units the point lies to the __________ of the origin. left or right The second number, –3, is called the ___________. y-coordinate (2, –3) It tells the number of units the point lies _____________ the origin. above or below

Name the points A, B, C, & D Point A(x, y) = (3, 2) B B A A
The Coordinate Plane Name the points A, B, C, & D y x 5 -4 -2 1 3 -5 -1 4 -3 2 Point A(x, y) = (3, 2) B (–3, 2) B A (3, 2) A Point B(x, y) = (–3, 2) Point C(x, y) = (–3, –2) Point D(x, y) = (3, –2) (–3, –2) C C D (3, –2) D

Consider these questions:
The Coordinate Plane On a piece of grid paper draw lines representing the x-axis and the y-axis. y x 5 -4 -2 1 3 -5 -1 4 -3 2 Graph : Point A(x, y) = A(2, 4) Point B(x, y) = B(2, 0) Point C(x, y) = C(2, –3) Point D(x, y) = D(2, –5) Consider these questions: x = 2 Write a general statement about ordered pairs that have the same x-coordinate. 1) What do you notice about the graphs of these points? They lie on a vertical line. They lie on a vertical line that intersects the x-axis at the x-coordinate. 2) What do you notice about the x-coordinates of these points? They are the same number.

On the same coordinate plane
The Coordinate Plane On the same coordinate plane y x 5 -4 -2 1 3 -5 -1 4 -3 2 Graph : Point W(x, y) = W(–4, –4) Point X(x, y) = X(–2, –4) Point Y(x, y) = Y(0, –4) Point Z(x, y) = Z(3, –4) y = – 4 Consider these questions: 1) What do you notice about the graphs of these points? Write a general statement about ordered pairs that have the same y-coordinate. They lie on a horizontal line. They lie on a horizontal line that intersects the y-axis at the y-coordinate. 2) What do you notice about the y-coordinates of these points? They are the same number.

If a and b are real numbers,
The Coordinate Plane Theorem 2 – 4 If a and b are real numbers, a vertical line contains all points (x, y) such that _____ and a horizontal line contains all points (x, y) such that _____ x = a y = b y x 5 -4 -2 1 3 -5 -1 4 -3 2 Graph the lines: x = –3 y = 2 (–3, 2) Graph the point of intersection of these lines.

The Coordinate Plane End of Lesson

1) Name the coordinates of each point.
5 Minute-Check y x 5 -1 2 4 -2 1 -3 3 1) Name the coordinates of each point. A = (2, 0) B = (3, –2) A B 2) Graph point C at (0, –4). y x 5 -4 -2 1 3 -5 -1 4 -3 2 x = –4 (–4, 2) 3) Graph x = –4. y = 2 4) Graph y = 2. C(0, -4) 5) Graph and label the intersection of x = – and y = 2

What You'll Learn Midpoints
You will learn to find the coordinates of the midpoint of a segment. B A C The midpoint of a line segment, , is the point C that ______ the segment. bisects A C B -7 -6 -5 -4 -2 -3 -1 1 2 3 4 5 6 7 C = [3 + (-5)] ÷ 2 = (-2) ÷ 2 = -1

Midpoints Theorem 2 – 5 On a number line, the coordinate of the midpoint of a segment whose endpoints have coordinates a and b is A B

Find the midpoint, C(x, y), of a segment on the coordinate plane.
Midpoints Find the midpoint, C(x, y), of a segment on the coordinate plane. Consider the x-coordinate: y x 10 -1 2 4 6 8 -2 3 7 1 5 9 x = 1 x = 9 It will be (midway) between the lines x = and x = 9 A B y = 7 C(x, y) Consider the y-coordinate: y x It will be (midway) between the lines y = and y = 7 y = 3

y x O On a coordinate plane, the coordinates of the midpoint of a
Midpoints Theorem 2 – 6 On a coordinate plane, the coordinates of the midpoint of a segment whose endpoints have coordinates (x1, y1) and (x2, y2) are O y x

Find the midpoint, C(x, y), of a segment on the coordinate plane.
Midpoints Find the midpoint, C(x, y), of a segment on the coordinate plane. y x 10 -1 2 4 6 8 -2 3 7 1 5 9 x = 1 x = 9 A(1, 7) B(9, 3) y = 7 C(5, 5) y x y = 3

Estimate the midpoint of AB.
Midpoints Graph A(1, 1) and B(7, 9) Draw AB y x 10 -1 2 4 6 8 -2 3 7 1 5 9 B(7, 9) Estimate the midpoint of AB. Check your answer using the midpoint formula. C C(4, 5) A(1, 1)

Suppose C(3, 5) is the midpoint of AB. Find the coordinate of B.
Midpoints Suppose C(3, 5) is the midpoint of AB. Find the coordinate of B. x-coordinate of B y-coordinate of B y x 10 -1 2 4 6 8 -2 3 7 1 5 9 B(-1, 8) Replace x1 with 7 and y1 with 2 B(x, y) is somewhere over there. C(3, 5) midpoint A(7, 2) Multiply each side by 2 Add or subtract to isolate the variable

Midpoints End of Lesson

A 2 1 4 3 6 5 7 9 8 10 2 1 4 3 6 5 7 9 8 10

1 2 3 1 2 4 3 5 7 6 8 9 10 11

Angles § 3.1 Angles § 3.2 Angle Measure
§ 3.3 The Angle Addition Postulate § 3.4 Adjacent Angles and Linear Pairs of Angles § 3.5 Complementary and Supplementary Angles § 3.6 Congruent Angles § 3.7 Perpendicular Lines 115

What You'll Learn Vocabulary
Angles What You'll Learn You will learn to name and identify parts of an angle. Vocabulary 1) Opposite Rays 2) Straight Angle 3) Angle 4) Vertex 5) Sides 6) Interior 7) Exterior 116

XY and XZ are ____________. opposite rays
Angles Opposite rays ___________ are two rays that are part of a the same line and have only their endpoints in common. X Y Z XY and XZ are ____________. opposite rays The figure formed by opposite rays is also referred to as a ____________. straight angle Straight Angle (Video) 117

There is another case where two rays can have a common endpoint.
Angles There is another case where two rays can have a common endpoint. This figure is called an _____. angle Some parts of angles have special names. S The common endpoint is called the ______, vertex and the two rays that make up the sides of the angle are called the sides of the angle. side R T side vertex 118

There are several ways to name this angle.
Angles There are several ways to name this angle. 1) Use the vertex and a point from each side. SRT or TRS S The vertex letter is always in the middle. 2) Use the vertex only. side R If there is only one angle at a vertex, then the angle can be named with that vertex. 1 R T side vertex 3) Use a number. 1 119

Angles Definition of Angle An angle is a figure formed by two noncollinear rays that have a common endpoint. E D F 2 Symbols: DEF FED E 2 Naming Angles (Video) 120

1) Name the angle in four ways.
Angles 1) Name the angle in four ways. B A 1 C ABC CBA B 1 2) Identify the vertex and sides of this angle. vertex: Point B sides: BA and BC 121

1) Name all angles having W as their vertex.
2 1 W 2 XWZ Y 2) What are other names for ? 1 Z XWY or YWX 3) Is there an angle that can be named ? W No! 122

An angle separates a plane into three parts:
1) the ______ interior exterior 2) the ______ exterior W Y Z A interior 3) the _________ angle itself In the figure shown, point B and all other points in the blue region are in the interior of the angle. B Point A and all other points in the green region are in the exterior of the angle. Points Y, W, and Z are on the angle. 123

Angles Is point B in the interior of the angle, exterior of the angle, or on the angle? P G Exterior B Is point G in the interior of the angle, exterior of the angle, or on the angle? On the angle Is point P in the interior of the angle, exterior of the angle, or on the angle? Interior 124

§3.2 Angle Measure What You'll Learn You will learn to measure, draw, and classify angles. Vocabulary 1) Degrees 2) Protractor 3) Right Angle 4) Acute Angle 5) Obtuse Angle 125

In geometry, angles are measured in units called _______. degrees
§3.2 Angle Measure In geometry, angles are measured in units called _______. degrees The symbol for degree is °. Q P R 75° In the figure to the right, the angle is 75 degrees. In notation, there is no degree symbol with 75 because the measure of an angle is a real number with no unit of measure. m PQR = 75 126

Angles Measure Postulate
§3.2 Angle Measure Postulate 3-1 Angles Measure Postulate For every angle, there is a unique positive number between __ and ____ called the degree measure of the angle. 180 B A C n° m ABC = n and 0 < n < 180 127

Use a protractor to measure SRQ.
§3.2 Angle Measure You can use a _________ to measure angles and sketch angles of given measure. protractor Use a protractor to measure SRQ. 1) Place the center point of the protractor on vertex R Align the straightedge with side RS. Q R S 2) Use the scale that begins with at RS Read where the other side of the angle, RQ, crosses this scale. 128

Find the measurement of: m SRH 70
§3.2 Angle Measure Find the measurement of: m SRH 70 m SRQ = 180 m QRG = 180 – 150 = 30 m SRJ = 45 m GRJ = 150 – 45 = 105 m SRG = 150 H J G S Q R 129

Use a protractor to draw an angle having a measure of 135.
§3.2 Angle Measure Use a protractor to draw an angle having a measure of 135. 1) Draw AB 3) Locate and draw point C at the mark labeled Draw AC. 2) Place the center point of the protractor on A. Align the mark labeled 0 with the ray. C A B 130

obtuse angle 90 < m A < 180 right angle m A = 90
§3.2 Angle Measure Once the measure of an angle is known, the angle can be classified as one of three types of angles. These types are defined in relation to a right angle. Types of Angles A A A obtuse angle 90 < m A < 180 right angle m A = 90 acute angle 0 < m A < 90 Angle Classification (Video) 131

Classify each angle as acute, obtuse, or right.
§3.2 Angle Measure Classify each angle as acute, obtuse, or right. 110° 90° 40° Obtuse Right Acute 75° 50° 130° Acute Obtuse Acute 132

The measure of B is 138. Solve for x.
§3.2 Angle Measure 5x - 7 B The measure of B is 138. Solve for x. 9y + 4 H The measure of H is 67. Solve for y. Given: (What do you know?) Given: (What do you know?) B = 5x – 7 and B = 138 H = 9y and H = 67 5x – 7 = 138 9y + 4 = 67 Check! Check! 5x = 145 9y = 63 5(29) -7 = ? 9(7) + 4 = ? x = 29 y = 7 = ? = ? 138 = 138 67 = 67 133

? ? ? Is m a larger than m b ? 60° 60° 134

End of Lesson

§3.3 The Angle Addition Postulate
What You'll Learn You will learn to find the measure of an angle and the bisector of an angle. Vocabulary NOTHING NEW! 136

1) Draw an acute, an obtuse, or a right angle Label the angle RST. R T S 45° 2) Draw and label a point X in the interior of the angle. Then draw SX. X 75° 30° 3) For each angle, find mRSX, mXST, and RST. 137

1) How does the sum of mRSX and mXST compare to mRST ? Their sum is equal to the measure of RST . mXST = 30 + mRSX = 45 = mRST = 75 R T S 2) Make a conjecture about the relationship between the two smaller angles and the larger angle. 45° X The sum of the measures of the two smaller angles is equal to the measure of the larger angle. The Angle Addition Postulate (Video) 75° 30° 138

§3.3 The Angle Addition Postulate Angle Addition Postulate
For any angle PQR, if A is in the interior of PQR, then mPQA + mAQR = mPQR. 2 1 A R P Q m1 + m2 = mPQR. There are two equations that can be derived using Postulate 3 – 3. m1 = mPQR – m2 These equations are true no matter where A is located in the interior of PQR. m2 = mPQR – m1 139

Find m2 if mXYZ = 86 and m1 = 22. 2 1 Y Z X W m2 + m1 = mXYZ Postulate 3 – 3. m2 = mXYZ – m1 m2 = 86 – 22 m2 = 64 140

Find mABC and mCBD if mABD = 120. mABC + mCBD = mABD Postulate 3 – 3. 2x + (5x – 6) = 120 Substitution 7x – 6 = 120 Combine like terms 7x = 126 Add 6 to both sides x = 18 Divide each side by 7 = 120 2x° (5x – 6)° B D C A mABC = 2x mCBD = 5x – 6 mABC = 2(18) mCBD = 5(18) – 6 mABC = 36 mCBD = 90 – 6 mCBD = 84 141

Just as every segment has a midpoint that bisects the segment, every angle has a ___ that bisects the angle. ray This ray is called an ____________ . angle bisector 142

Definition of an Angle Bisector The bisector of an angle is the ray with its endpoint at the vertex of the angle, extending into the interior of the angle. The bisector separates the angle into two angles of equal measure. 2 1 A R P Q is the bisector of PQR. m1 = m2 143

If bisects CAN and mCAN = 130, find 1 and 2. Since bisects CAN, 1 = 2. 1 2 A C N T 1 + 2 = CAN Postulate 3 - 3 1 + 2 = 130 Replace CAN with 130 1 + 1 = 130 Replace 2 with 1 2(1) = 130 Combine like terms (1) = 65 Divide each side by 2 Since 1 = 2, 2 = 65 144

End of Lesson

Adjacent Angles and Linear Pairs of Angles
What You'll Learn You will learn to identify and use adjacent angles and linear pairs of angles. A C B When you “split” an angle, you create two angles. The two angles are called _____________ adjacent angles D 2 1 adjacent = next to, joining. 1 and 2 are examples of adjacent angles. They share a common ray. Name the ray that 1 and 2 have in common. ____ 146

Definition of Adjacent Angles Adjacent angles are angles that: A) share a common side B) have the same vertex, and C) have no interior points in common M J N R 1 2 1 and 2 are adjacent with the same vertex R and common side 147

Determine whether 1 and 2 are adjacent angles. No. They have a common vertex B, but _____________ 1 2 B no common side 1 2 G Yes. They have the same vertex G and a common side with no interior points in common. N 1 2 J L No. They do not have a common vertex or ____________ a common side The side of 1 is ____ The side of 2 is ____ 148

Determine whether 1 and 2 are adjacent angles. No. 1 2 1 2 X D Z Yes. In this example, the noncommon sides of the adjacent angles form a ___________. straight line These angles are called a _________ linear pair 149

Definition of Linear Pairs Two angles form a linear pair if and only if (iff): A) they are adjacent and B) their noncommon sides are opposite rays C A D B 1 2 1 and 2 are a linear pair. Note: 150

In the figure, and are opposite rays. 1 2 M 4 3 E H T A C 1) Name the angle that forms a linear pair with 1. ACE ACE and 1 have a common side , the same vertex C, and opposite rays and 2) Do 3 and TCM form a linear pair? Justify your answer. No. Their noncommon sides are not opposite rays. 151

End of Lesson

§3.5 Complementary and Supplementary Angles
What You'll Learn You will learn to identify and use Complementary and Supplementary angles 153

Definition of Complementary Angles Two angles are complementary if and only if (iff) the sum of their degree measure is 90. 60° D E F 30° A B C mABC + mDEF = = 90 154

If two angles are complementary, each angle is a complement of the other. ABC is the complement of DEF and DEF is the complement of ABC. 60° D E F 30° A B C Complementary angles DO NOT need to have a common side or even the same vertex. 155

Some examples of complementary angles are shown below. 75° I 15° H mH + mI = 90 50° H 40° Q P S mPHQ + mQHS = 90 30° 60° T U V W Z mTZU + mVZW = 90 156

If the sum of the measure of two angles is 180, they form a special pair of angles called supplementary angles. Definition of Supplementary Angles Two angles are supplementary if and only if (iff) the sum of their degree measure is 180. 130° D E F 50° A B C mABC + mDEF = = 180 157

Some examples of supplementary angles are shown below. 105° H 75° I mH + mI = 180 50° H 130° Q P S mPHQ + mQHS = 180 60° 120° T U V W Z mTZU + mUZV = 180 and mTZU + mVZW = 180 158

End of Lesson

What You'll Learn You will learn to identify and use congruent and
§3.6 Congruent Angles What You'll Learn You will learn to identify and use congruent and vertical angles. Recall that congruent segments have the same ________. measure Congruent angles _______________ also have the same measure. 160

Two angles are congruent iff, they have the same ______________.
§3.6 Congruent Angles Definition of Congruent Angles Two angles are congruent iff, they have the same ______________. degree measure B  V iff 50° V 50° B mB = mV 161

To show that 1 is congruent to 2, we use ____. arcs
§3.6 Congruent Angles To show that 1 is congruent to 2, we use ____. arcs 2 1 To show that there is a second set of congruent angles, X and Z, we use double arcs. This “arc” notation states that: Z X X  Z mX = mZ 162

When two lines intersect, ____ angles are formed. four
§3.6 Congruent Angles When two lines intersect, ____ angles are formed. four There are two pair of nonadjacent angles. These pairs are called _____________. vertical angles 1 4 2 3 163

Two angles are vertical iff they are two nonadjacent
§3.6 Congruent Angles Definition of Vertical Angles Two angles are vertical iff they are two nonadjacent angles formed by a pair of intersecting lines. Vertical angles: 1 and 3 1 4 2 2 and 4 3 164

Hands-On 1) On a sheet of paper, construct two intersecting lines
§3.6 Congruent Angles 1) On a sheet of paper, construct two intersecting lines that are not perpendicular. 2) With a protractor, measure each angle formed. 3) Make a conjecture about vertical angles. 1 2 3 4 Consider: A. 1 is supplementary to 4. m1 + m4 = 180 Hands-On B. 3 is supplementary to 4. m3 + m4 = 180 Therefore, it can be shown that 1  3 Likewise, it can be shown that 2  4 165

1) If m1 = 4x + 3 and the m3 = 2x + 11, then find the m3
§3.6 Congruent Angles 1 2 3 4 1) If m1 = 4x + 3 and the m3 = 2x + 11, then find the m3 x = 4; 3 = 19° 2) If m2 = x + 9 and the m3 = 2x + 3, then find the m4 x = 56; 4 = 65° 3) If m2 = 6x - 1 and the m4 = 4x + 17, then find the m3 x = 9; 3 = 127° 4) If m1 = 9x - 7 and the m3 = 6x + 23, then find the m4 x = 10; 4 = 97° 166

Vertical angles are congruent.
§3.6 Congruent Angles Theorem 3-1 Vertical Angle Theorem Vertical angles are congruent. n m 2 1  3 3 1 2  4 4 167

Find the value of x in the figure:
§3.6 Congruent Angles Find the value of x in the figure: The angles are vertical angles. 130° x° So, the value of x is 130°. 168

Find the value of x in the figure:
§3.6 Congruent Angles Find the value of x in the figure: The angles are vertical angles. (x – 10) = 125. (x – 10)° x – 10 = 125. 125° x = 135. 169

Suppose two angles are congruent.
§3.6 Congruent Angles Suppose two angles are congruent. What do you think is true about their complements? 1  2 1 + x = 90 2 + y = 90 x is the complement of 1 y is the complement of 2 x = 90 - 1 y = 90 - 2 Because 1  2, a “substitution” is made. x = 90 - 1 y = 90 - 1 x = y x  y If two angles are congruent, their complements are congruent. 170

If two angles are congruent, then their complements are _________.
§3.6 Congruent Angles Theorem 3-2 If two angles are congruent, then their complements are _________. congruent The measure of angles complementary to A and B is 30. A B 60° A  B Theorem 3-3 If two angles are congruent, then their supplements are _________. congruent The measure of angles supplementary to 1 and 4 is 110. 4 3 2 1 70° 110° 110° 70° 4  1 171

If two angles are complementary to the same angle,
§3.6 Congruent Angles Theorem 3-4 If two angles are complementary to the same angle, then they are _________. congruent 3 is complementary to 4 5 is complementary to 4 4 3 5  3 5 Theorem 3-5 If two angles are supplementary to the same angle, then they are _________. congruent 3 1 2 1 is supplementary to 2 3 is supplementary to 2 1  3 172

Find the measure of an angle that is supplementary to B.
§3.6 Congruent Angles Suppose A  B and mA = 52. Find the measure of an angle that is supplementary to B. B 52° A 52° 1 B + 1 = 180 1 = 180 – B 1 = 180 – 52 1 = 128° 173

If 1 is complementary to 3, 2 is complementary to 3, and m3 = 25,
§3.6 Congruent Angles If 1 is complementary to 3, 2 is complementary to 3, and m3 = 25, What are m1 and m2 ? m1 + m3 = Definition of complementary angles. m1 = 90 - m Subtract m3 from both sides. m1 = Substitute 25 in for m3. m1 = Simplify the right side. You solve for m2 m2 + m3 = Definition of complementary angles. m2 = 90 - m Subtract m3 from both sides. m2 = Substitute 25 in for m3. m2 = Simplify the right side. 174

1) If m1 = 2x + 3 and the m3 = 3x - 14, then find the m3
§3.6 Congruent Angles A B C D E G H 1 2 3 4 1) If m1 = 2x + 3 and the m3 = 3x - 14, then find the m3 x = 17; 3 = 37° 2) If mABD = 4x + 5 and the mDBC = 2x + 1, then find the mEBC x = 29; EBC = 121° 3) If m1 = 4x and the m3 = 2x + 19, then find the m4 x = 16; 4 = 39° 4) If mEBG = 7x and the mEBH = 2x + 7, then find the m1 x = 18; 1 = 43° 175

Suppose you draw two angles that are congruent and supplementary.
What is true about the angles? 176

All right angles are _________. congruent
§3.6 Congruent Angles Theorem 3-6 If two angles are congruent and supplementary then each is a __________. right angle 1 is supplementary to 2 1 2 1 and 2 = 90 Theorem 3-7 All right angles are _________. congruent C B A A  B  C 177

If 1 is supplementary to 4, 3 is supplementary to 4, and
§3.6 Congruent Angles If 1 is supplementary to 4, 3 is supplementary to 4, and m 1 = 64, what are m 3 and m 4? A D C B E 1 2 3 4 1  3 They are vertical angles. m 1 = m3 m 3 = 64 3 is supplementary to 4 Given m3 + m4 = 180 Definition of supplementary. 64 + m4 = 180 m4 = 180 – 64 m4 = 116 178

End of Lesson

§3.7 Perpendicular Lines What You'll Learn You will learn to identify, use properties of, and construct perpendicular lines and segments. 180

Lines that intersect at an angle of 90 degrees are _________________.
§3.7 Perpendicular Lines Lines that intersect at an angle of 90 degrees are _________________. perpendicular lines In the figure below, lines are perpendicular. A D C B 1 2 3 4 181

Perpendicular lines are lines that intersect to form a right angle.
Definition of Perpendicular Lines Perpendicular lines are lines that intersect to form a right angle. m n 182

In the figure below, l  m. The following statements are true.
§3.7 Perpendicular Lines In the figure below, l  m. The following statements are true. m 2 1 3 4 l 1) 1 is a right angle. Definition of Perpendicular Lines 2) 1  3. Vertical angles are congruent 3) 1 and 4 form a linear pair. Definition of Linear Pair 4) 1 and 4 are supplementary. Linear pairs are supplementary 5) 4 is a right angle. m = 180, m4 = 90 6) 2 is a right angle. Vertical angles are congruent 183

If two lines are perpendicular, then they form four right angles.
§3.7 Perpendicular Lines Theorem 3-8 If two lines are perpendicular, then they form four right angles. 1 3 4 2 a b 184

1) PRN is an acute angle. False. 2) 4  8 True
§3.7 Perpendicular Lines 1) PRN is an acute angle. False. 2) 4  8 True 185

If a line m is in a plane and T is a point in m, then there
§3.7 Perpendicular Lines Theorem 3-9 If a line m is in a plane and T is a point in m, then there exists exactly ___ line in that plane that is perpendicular to m at T. one m T 186

End of Lesson

Parallels § 4.1 Parallel Lines and Planes
§ 4.2 Parallel Lines and Transversals § 4.3 Transversals and Corresponding Angles § 4.4 Proving Lines Parallel § 4.5 Slope § 4.6 Equations of Lines

Parallel Lines and Planes
What You'll Learn You will learn to describe relationships among lines, parts of lines, and planes. In geometry, two lines in a plane that are always the same distance apart are ____________. parallel lines No two parallel lines intersect, no matter how far you extend them.

Definition of Parallel Lines Two lines are parallel iff they are in the same plane and do not ________. intersect

Planes can also be parallel. The shelves of a bookcase are examples of parts of planes. The shelves are the same distance apart at all points, and do not appear to intersect. They are _______. parallel In geometry, planes that do not intersect are called _____________. parallel planes Q J K M L S R P Plane PSR || plane JML Plane JPQ || plane MLR Plane PJM || plane QRL

Sometimes lines that do not intersect are not in the same plane. These lines are called __________. skew lines Definition of Skew Lines Two lines that are not in the same plane are skew iff they do not intersect.

Name the parts of the figure: 1) All planes parallel to plane ABF Plane DCG 2) All segments that intersect A C B E G H D F AD, CD, GH, AH, EH 3) All segments parallel to AB, GH, EF 4) All segments skew to DH, CG, FG, EH

End of Section 4.1

Parallel Lines and Transversals
What You'll Learn You will learn to identify the relationships among pairs of interior and exterior angles formed by two parallel lines and a transversal.

In geometry, a line, line segment, or ray that intersects two or more lines at different points is called a __________ transversal B A l m 1 2 4 3 5 6 8 7 is an example of a transversal. It intercepts lines l and m. Note all of the different angles formed at the points of intersection.

Definition of Transversal In a plane, a line is a transversal iff it intersects two or more Lines, each at a different point. The lines cut by a transversal may or may not be parallel. l m 1 2 3 4 5 7 6 8 Parallel Lines t is a transversal for l and m. t 1 2 3 4 5 7 6 8 b c Nonparallel Lines r is a transversal for b and c. r

Two lines divide the plane into three regions. The region between the lines is referred to as the interior. The two regions not between the lines is referred to as the exterior. Exterior Interior

When a transversal intersects two lines, _____ angles are formed. eight These angles are given special names. l m 1 2 3 4 5 7 6 8 t Interior angles lie between the two lines. Exterior angles lie outside the two lines. Alternate Interior angles are on the opposite sides of the transversal. Alternate Exterior angles are on the opposite sides of the transversal. Consectutive Interior angles are on the same side of the transversal. ?

Theorem 4-1 Alternate Interior Angles If two parallel lines are cut by a transversal, then each pair of Alternate interior angles is _________. congruent 1 2 4 3 5 6 8 7

Theorem 4-2 Consecutive Interior Angles If two parallel lines are cut by a transversal, then each pair of consecutive interior angles is _____________. supplementary 1 2 3 4 5 7 6 8

Theorem 4-3 Alternate Exterior Angles If two parallel lines are cut by a transversal, then each pair of alternate exterior angles is _________. congruent 1 2 3 4 5 7 6 8 ?

End of Lesson Practice Problems:
1, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, and (total = 23)

Transversals and Corresponding Angles
What You'll Learn You will learn to identify the relationships among pairs of corresponding angles formed by two parallel lines and a transversal.

When a transversal crosses two lines, the intersection creates a number of angles that are related to each other. Note 1 and 5 below. Although one is an exterior angle and the other is an interior angle, both lie on the same side of the transversal. Angle 1 and 5 are called __________________. corresponding angles l m 1 2 3 4 5 7 6 8 t Give three other pairs of corresponding angles that are formed: 4 and 8 3 and 7 2 and 6

Postulate 4-1 Corresponding Angles If two parallel lines are cut by a transversal, then each pair of corresponding angles is _________. congruent

Concept Summary Congruent Supplementary Types of angle pairs formed when a transversal cuts two parallel lines. alternate interior consecutive interior alternate exterior corresponding

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 s || t and c || d. Name all the angles that are congruent to 1. Give a reason for each answer. 3  1 corresponding angles 6  1 vertical angles 8  1 alternate exterior angles 9  1 corresponding angles 14  1 alternate exterior angles 11  9  1 corresponding angles 16  14  1 corresponding angles

End of Lesson Practice Problems: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20,
22, 24, 26, 28, 30, 32, 34, 36, and 38 (total = 19)

Proving Lines Parallel
What You'll Learn You will learn to identify conditions that produce parallel lines. Reminder: In Chapter 1, we discussed “if-then” statements (pg. 24). Within those statements, we identified the “__________” and the “_________”. hypothesis conclusion I said then that in mathematics, we only use the term “if and only if” if the converse of the statement is true.

Postulate 4 – 1 (pg. 156): IF ___________________________________, THEN ________________________________________. two parallel lines are cut by a transversal two parallel lines are cut by a transversal each pair of corresponding angles is congruent each pair of corresponding angles is congruent The postulates used in §4 - 4 are the converse of postulates that you already know. COOL, HUH? §4 – 4, Postulate 4 – 2 (pg. 162): IF ________________________________________, THEN ____________________________________.

Postulate 4-2 In a plane, if two lines are cut by a transversal so that a pair of corresponding angles is congruent, then the lines are _______. parallel 1 2 a b If 1 2, then _____ a || b

Theorem 4-5 In a plane, if two lines are cut by a transversal so that a pair of alternate interior angles is congruent, then the two lines are _______. parallel 1 2 a b If 1 2, then _____ a || b

Theorem 4-6 In a plane, if two lines are cut by a transversal so that a pair of alternate exterior angles is congruent, then the two lines are _______. parallel 1 2 a b If 1 2, then _____ a || b

Theorem 4-7 In a plane, if two lines are cut by a transversal so that a pair of consecutive interior angles is supplementary, then the two lines are _______. parallel 1 2 a b If 1 + 2 = 180, then _____ a || b

Theorem 4-8 In a plane, if two lines are cut by a transversal so that a pair of consecutive interior angles is supplementary, then the two lines are _______. parallel If a  t and b  t, then _____ a b t a || b

We now have five ways to prove that two lines are parallel. Concept Summary Show that a pair of corresponding angles is congruent. Show that a pair of alternate interior angles is congruent. Show that a pair of alternate exterior angles is congruent. Show that a pair of consecutive interior angles is supplementary. Show that two lines in a plane are perpendicular to a third line.

Identify any parallel segments. Explain your reasoning. G A Y D R 90°

Find the value for x so BE || TS. E B S T (6x - 26)° (2x + 10)° (5x + 2)° mBES + mEST = 180 ES is a transversal for BE and TS. BES and EST are _________________ angles. (2x + 10) + (5x + 2) = 180 consecutive interior 7x = 180 If mBES + mEST = 180, then BE || TS by Theorem 4 – 7. 7x = 168 x = 24 Thus, if x = 24, then BE || TS.

End of Lesson Practice Problems: 1, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14,
15, 16, 17, 18, 19, 21, 25, and 26 (total = 19)

Slope What You'll Learn You will learn to find the slopes of lines and use slope to identify parallel and perpendicular lines.

If the pilot doesn’t change something, he / she will not make it home for Christmas. Would you agree? Consider the options: There has got to be some “measurable” way to get this aircraft to clear such obstacles. Discuss how you might radio a pilot and tell him or her how to adjust the slope of their flight path in order to clear the mountain. 1) Keep the same slope of his / her path. Not a good choice! 2) Go straight up. Not possible! This is an airplane, not a helicopter.

Fortunately, there is a way to measure a proper “slope” to clear the obstacle.
We measure the “change in height” required and divide that by the “horizontal change” required.

y x 10000

The steepness of a line is called the _____. slope
Slope is defined as the ratio of the ____, or vertical change, to the ___, or horizontal change, as you move from one point on the line to another. rise run y x 10 -5 -10 5

The slope m of the non-vertical line passing through the points and is
y x

The slope “m” of a line containing two points with coordinates
Definition of Slope The slope “m” of a line containing two points with coordinates (x1, y1), and (x2, y2), is given by the formula

Slope The slope m of a non-vertical line is the number of units the line rises or falls for each unit of horizontal change from left to right. y x (3, 6) rise = = 5 units (1, 1) run = = 2 units ? 6 & 7

Slope Postulate 4 – 3 Two distinct nonvertical lines are parallel iff they have _____________. the same slope

Slope Postulate 4 – 4 Two nonvertical lines are perpendicular iff ___________________________. the product of their slope is -1 ? 8 & 9

End of Lesson Practice Problems: 1, 3, 4, 5, 6, 7, 8, 9, 10, 12,
14, 16, 17, 20, 22, 24, 26, 30, and 32 (total = 19)

Equations of Lines What You'll Learn You will learn to write and graph equations of lines. The equation y = 2x – 1 is called a _____________ because its graph is a straight line. linear equation We can substitute different values for x in the graph to find corresponding values for y. y x 8 1 3 5 7 -1 2 4 6 x y = 2x -1 y There are many more points whose ordered pairs are solutions of y = 2x – 1. These points also lie on the line. (3, 5) 1 y = 2(1) -1 1 2 (2, 3) y = 2(2) -1 3 3 y = 2(3) -1 5 (1, 1)

y = mx + b Look at the graph of y = 2x – 1 .
Equations of Lines Look at the graph of y = 2x – 1 . The y – value of the point where the line crosses the y-axis is ___. - 1 This value is called the ____________ of the line. y - intercept Most linear equations can be written in the form __________. y = mx + b This form is called the ___________________. slope – intercept form y x 5 -2 1 3 -3 2 -1 4 y = 2x – 1 y = mx + b slope y - intercept (0, -1)

An equation of the line having slope m and y-intercept b is y = mx + b
Equations of Lines Slope – Intercept Form An equation of the line having slope m and y-intercept b is y = mx + b

1) Rewrite the equation in slope – intercept form by solving for y.
Equations of Lines 1) Rewrite the equation in slope – intercept form by solving for y. 2x – 3 y = 18 2) Graph 2x + y = 3 using the slope and y – intercept. y = –2x + 3 y x 5 -2 1 3 -3 2 -1 4 1) Identify and graph the y-intercept. (0, 3) 2) Follow the slope a second point on the line. (1, 1) 3) Draw the line between the two points.

3) Write an equation of the line perpendicualr to the graph of
Equations of Lines 1) Write an equation of the line parallel to the graph of y = 2x – 5 that passes through the point (3, 7). y = 2x + 1 2) Write an equation of the line parallel to the graph of 3x + y = 6 that passes through the point (1, 4). y = -3x + 7 3) Write an equation of the line perpendicualr to the graph of that passes through the point ( - 3, 8). y = -4x -4

End of Lesson Practice Problems:
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 40, and (total = 24)

Triangles and Congruence
§ 5.1 Classifying Triangles § 5.2 Angles of a Triangle § 5.3 Geometry in Motion § 5.4 Congruent Triangles § 5.5 SSS and SAS § 5.6 ASA and AAS

Classifying Triangles
What You'll Learn You will learn to identify the parts of triangles and to classify triangles by their parts. In geometry, a triangle is a figure formed when _____ noncollinear points are connected by segments. three E D F Each pair of segments forms an angle of the triangle. The vertex of each angle is a vertex of the triangle.

Triangles are named by the letters at their vertices. Triangle DEF, written ______, is shown below. ΔDEF vertex The sides are: EF, FD, and DE. E D F angle The vertices are: D, E, and F. The angles are: E, F, and D. side In Chapter 3, you classified angles as acute, obtuse, or right. Triangles can also be classified by their angles. All triangles have at least two _____ angles. acute The third angle is either _____, ______, or _____. acute obtuse right

Classified by Angles acute triangle obtuse triangle right triangle 30° 60° 60° 80° 120° 17° 43° 40° 3rd angle is _____ 3rd angle is ______ 3rd angle is ____ acute obtuse right

Classified by Sides scalene isosceles equilateral ___ sides congruent no at least two __________ sides congruent ___ sides congruent all

The angle formed by the congruent sides is called the ___________. vertex angle The two angles formed by the base and one of the congruent sides are called ___________. The congruent sides are called legs. base angles leg leg The side opposite the vertex angle is called the _____. base

End of Section 5.1

What You'll Learn You will learn to use the Angle Sum Theorem.
Angles of a Triangle What You'll Learn You will learn to use the Angle Sum Theorem. 1) On a piece of paper, draw a triangle. 2) Place a dot close to the center (interior) of the triangle. 3) After marking all of the angles, tear the triangle into three pieces. then rotate them, laying the marked angles next to each other. 4) Make a conjecture about the sum of the angle measures of the triangle.

The sum of the measures of the angles of a triangle is 180.
Theorem 5-1 Angle Sum Theorem The sum of the measures of the angles of a triangle is 180. z° x° y° x + y + z = 180

The acute angles of a right triangle are complementary.
Angles of a Triangle Theorem 5-2 The acute angles of a right triangle are complementary. x° y° x + y = 90

The measure of each angle of an equiangular triangle is 60.
Angles of a Triangle Theorem 5-3 The measure of each angle of an equiangular triangle is 60. x° 3x = 180 x = 60

End of Section 5.2

Geometry in Motion What You'll Learn You will learn to identify translations, reflections, and rotations and their corresponding parts. We live in a world of motion. Geometry helps us define and describe that motion. In geometry, there are three fundamental types of motion: __________, _________, and ________. translation reflection rotation

Geometry in Motion Translation In a translation, you slide a figure from one position to another without turning it. Translations are sometimes called ______. slides

Reflection In a reflection, you flip a figure over a line.
Geometry in Motion line of reflection Reflection In a reflection, you flip a figure over a line. The new figure is a mirror image. Reflections are sometimes called ____. flips

Rotation In a rotation, you rotate a figure around a fixed point.
Geometry in Motion Rotation In a rotation, you rotate a figure around a fixed point. Rotations are sometimes called _____. turns 30°

This one-to-one correspondence is an example of a _______. mapping
Geometry in Motion A B C D E F Each point on the original figure is called a _________. Its matching point on the corresponding figure is called its ______. preimage image Each point on the preimage can be paired with exactly one point on its image, and each point on the image can be paired with exactly one point on its preimage. This one-to-one correspondence is an example of a _______. mapping

The symbol → is used to indicate a mapping.
Geometry in Motion A B C D E F Each point on the original figure is called a _________. Its matching point on the corresponding figure is called its ______. preimage image The symbol → is used to indicate a mapping. In the figure, ΔABC → ΔDEF (ΔABC maps to ΔDEF). In naming the triangles, the order of the vertices indicates the corresponding points.

→ → → → → → A D AB DE B E BC EF C F CA FD Each point on Its matching
Geometry in Motion A B C D E F Each point on the original figure is called a _________. Its matching point on the corresponding figure is called its ______. preimage image Preimage Image Preimage Image → A D → AB DE → B E BC → EF → C F → CA FD This mapping is called a _____________. transformation

Translations, reflections, and rotations are all __________.
Geometry in Motion Translations, reflections, and rotations are all __________. isometries An isometry is a movement that does not change the size or shape of the figure being moved. When a figure is translated, reflected, or rotated, the lengths of the sides of the figure DO NOT CHANGE.

End of Section 5.3

What You'll Learn ΔABC  ΔXYZ
Congruent Triangles What You'll Learn You will learn to identify corresponding parts of congruent triangles If a triangle can be translated, rotated, or reflected onto another triangle, so that all of the vertices correspond, the triangles are _________________. congruent triangles The parts of congruent triangles that “match” are called __________________. corresponding parts The order of the ________ indicates the corresponding parts! ΔABC  ΔXYZ vertices

These relationships help define the congruent triangles.
In the figure, ΔABC  ΔFDE. A As in a mapping, the order of the _______ indicates the corresponding parts. vertices C B Congruent Angles Congruent Sides A  F AB  FD F E D B  D BC  DE C  E AC  FE These relationships help define the congruent triangles.

If the _________________ of two triangles are congruent, then
Congruent Triangles Definition of Congruent Triangles If the _________________ of two triangles are congruent, then the two triangles are congruent. corresponding parts If two triangles are _________, then the corresponding parts of the two triangles are congruent. congruent

ΔRST  ΔXYZ ΔRST  ΔXYZ. Find the value of n.
Congruent Triangles ΔRST  ΔXYZ. Find the value of n. T S R Z X Y 40° (2n + 10)° 50° 90° ΔRST  ΔXYZ identify the corresponding parts corresponding parts are congruent S  Y 50 = 2n + 10 subtract 10 from both sides 40 = 2n divide both sides by 2 20 = n

End of Section 5.4

SSS and SAS What You'll Learn You will learn to use the SSS and SAS tests for congruency.

4) Construct a segment congruent to CB.
SSS and SAS 2) Construct a segment congruent to AC. Label the endpoints of the segment D and E. 4) Construct a segment congruent to CB. 3) Construct a segment congruent to AB. 1) Draw an acute scalene triangle on a piece of paper. Label its vertices A, B, and C, on the interior of each angle. 5) Label the intersection F. 6) Draw DF and EF. A C B D E F This activity suggests the following postulate.

Triangles are congruent. sides three corresponding
SSS and SAS Postulate 5-1 SSS Postulate If three _____ of one triangle are congruent to _____ _____________ sides of another triangle, then the two Triangles are congruent. sides three corresponding A B C R S T If AC  RT and AB  RS and BC  ST then ΔABC  ΔRST

In two triangles, ZY  FE, XY  DE, and XZ  DF.
SSS and SAS In two triangles, ZY  FE, XY  DE, and XZ  DF. Write a congruence statement for the two triangles. X D Z Y F E Sample Answer: ΔZXY  ΔFDE

In a triangle, the angle formed by two given sides is called the
SSS and SAS In a triangle, the angle formed by two given sides is called the ____________ of the sides. included angle C is the included angle of CA and CB A B C A is the included angle of AB and AC B is the included angle of BA and BC Using the SSS Postulate, you can show that two triangles are congruent if their corresponding sides are congruent. You can also show their congruence by using two sides and the ____________. included angle

If ________ and the ____________ of one triangle are
SSS and SAS Postulate 5-2 SAS Postulate If ________ and the ____________ of one triangle are congruent to the corresponding sides and included angle of another triangle, then the triangles are congruent. two sides included angle A B C R S T If AC  RT and A  R and AB  RS then ΔABC  ΔRST

NO! On a piece of paper, write your response to the following:
SSS and SAS On a piece of paper, write your response to the following: Determine whether the triangles are congruent by SAS. If so, write a statement of congruence and tell why they are congruent. If not, explain your reasoning. P R Q F E D NO! D is not the included angle for DF and EF.

End of Section 5.5

ASA and AAS What You'll Learn You will learn to use the ASA and AAS tests for congruency.

It is the one side common to both angles.
ASA and AAS The side of a triangle that falls between two given angles is called the ___________ of the angles. included side It is the one side common to both angles. A B C AC is the included side of A and C CB is the included side of C and B AB is the included side of A and B You can show that two triangles are congruent by using _________ and the ___________ of the triangles. two angles included side

If _________ and the ___________ of one triangle are
ASA and AAS Postulate 5-3 ASA Postulate If _________ and the ___________ of one triangle are congruent to the corresponding angles and included side of another triangle, then the triangles are congruent. two angles included side A B C R S T If A  R and AC  RT and C  T then ΔABC  ΔRST

CA and CB are the nonincluded
ASA and AAS CA and CB are the nonincluded sides of A and B A B C You can show that two triangles are congruent by using _________ and a ______________. two angles nonincluded side

If _________ and a ______________ of one triangle are
ASA and AAS Theorem 5-4 AAS Theorem If _________ and a ______________ of one triangle are congruent to the corresponding two angles and nonincluded side of another triangle, then the triangles are congruent. two angles nonincluded side A B C R S T If A  R and C  T and CB  TS then ΔABC  ΔRST

ΔDEF and ΔLNM have one pair of sides and one pair of angles marked to
ASA and AAS ΔDEF and ΔLNM have one pair of sides and one pair of angles marked to show congruence. What other pair of angles must be marked so that the two triangles are congruent by AAS? If F and M are marked congruent, then FE and MN would be included sides. However, AAS requires the nonincluded sides. Therefore, D and L must be marked. D F E L M N

End of Section 5.6

More About Triangles § 6.1 Medians
§ 6.2 Altitudes and Perpendicular Bisectors § 6.3 Angle Bisectors of Triangles § 6.4 Isosceles Triangles § 6.5 Right Triangles § 6.6 The Pythagorean Theorem § 6.7 Distance on the Coordinate Plane

Medians What You'll Learn You will learn to identify and construct medians in triangles 1) ______ 2) _______ 3) _________ Vocabulary median centroid concurrent

the ________ of the side __________________. vertex midpoint
Medians In a triangle, a median is a segment that joins a ______ of the triangle and the ________ of the side __________________. vertex midpoint opposite that vertex C B A D F E The medians of ΔABC, AD, BE, and CF, intersect at a common point called the ________. centroid When three or more lines or segments meet at the same point, the lines are __________. concurrent

and the length of the segment from the centroid to the midpoint.
Medians There is a special relationship between the length of the segment from the vertex to the centroid and the length of the segment from the centroid to the midpoint. C B A E F D

The length of the segment from the vertex to the centroid is
Medians Theorem 6 - 1 The length of the segment from the vertex to the centroid is _____ the length of the segment from the centroid to the midpoint. twice 2x x When three or more lines or segments meet at the same point, the lines are __________. concurrent

Medians CD = 14 D C B A E F

End of Section 6.1

Altitudes and Perpendicular Bisectors
What You'll Learn You will learn to identify and construct _______ and __________________ in triangles. altitudes perpendicular bisectors 1) ______ 2) __________________ Vocabulary altitude perpendicular bisector

In geometry, an altitude of a triangle is a ____________ segment with one endpoint at a ______ and the other endpoint on the side _______ that vertex. perpendicular vertex opposite C A B D The altitude AD is perpendicular to side BC.

Constructing an altitude of a triangle 3) Place the compass point at D and draw an arc below AC. Using the same compass setting, place the compass point on E and draw an arc to intersect the one drawn. 2) Place the compass point on B and draw an arc that intersects side AC in two points. Label the points of intersection D and E. 1) Draw a triangle like ΔABC 4) Use a straightedge to align the vertex B and the point where the two arcs intersect. Draw a segment from the vertex B to side AC. Label the point of intersection F. B A C D E

An altitude of a triangle may not always lie inside the triangle. Altitudes of Triangles acute triangle right triangle obtuse triangle The altitude is _____ the triangle The altitude is _____ of the triangle The altitude is _______ the triangle inside a side out side

Another special line in a triangle is a perpendicular bisector. A perpendicular line or segment that bisects a ____ of a triangle is called the perpendicular bisector of that side. side Line m is the perpendicular bisector of side BC. m A altitude B D C D is the midpoint of BC.

In some triangles, the perpendicular bisector and the altitude are the same. The line containing YE is the perpendicular bisector of XZ. Y E YE is an altitude. E is the midpoint of XZ. X Z

End of Section 6.2

Angle Bisectors of Triangles
What You'll Learn You will learn to identify and use ____________ in triangles. angle bisectors 1) ___________ Vocabulary angle bisector

Recall that the bisector of an angle is a ray that separates the angle into two congruent angles. Q R P S

An angle bisector of a triangle is a segment that separates an angle of the triangle into two congruent angles. One of the endpoints of an angle bisector is a ______ of the triangle, vertex and the other endpoint is on the side ________ that vertex. opposite A B D C Just as every triangle has three medians, three altitudes, and three perpendicular bisectors, every triangle has three angle bisectors.

Special Segments in Triangles Segment Type Property altitude perpendicular bisector angle bisector line segment line ray line segment line segment from the vertex, a line perpendicular to the opposite side bisects the side of the triangle bisects the angle of the triangle

End of Section 6.3

Isosceles Triangles What You'll Learn You will learn to identify and use properties of _______ triangles. isosceles 1) _____________ 2) ____ 3) ____ Vocabulary isosceles triangle base legs

The congruent sides are called ____. legs
Isosceles Triangles Recall from §5-1 that an isosceles triangle has at least two congruent sides. The congruent sides are called ____. legs The side opposite the vertex angle is called the ____. base In an isosceles triangle, there are two base angles, the vertices where the base intersects the congruent sides. vertex angle leg base angle base

triangle are congruent, then the angles opposite those sides
Isosceles Triangles Theorem 6-2 Isosceles Triangle 6-3 A B C If two sides of a triangle are congruent, then the angles opposite those sides are congruent. The median from the vertex angle of an isosceles triangle lies on the perpendicular bisector of the base and the angle bisector of the vertex angle. A B C D

triangle are congruent, then the sides opposite those angles
Isosceles Triangles Theorem 6-4 Converse of Isosceles Triangle A B C If two angles of a triangle are congruent, then the sides opposite those angles are congruent. Theorem 6-5 A triangle is equilateral if and only if it is equiangular.

End of Section 6.4

Right Triangles What You'll Learn You will learn to use tests for _________ of ____ triangles. congruence right 1) _________ 2) ____ Vocabulary hypotenuse legs

In a right triangle, the side opposite the right angle is called the
Right Triangles In a right triangle, the side opposite the right angle is called the _________. hypotenuse The two sides that form the right angle are called the ____. legs hypotenuse leg leg

Right Triangles Recall from Chapter 5, we studied various ways to prove triangles to be congruent: In §5-5, we studied two theorems A B C R S T SSS and A B C R S T SAS

Right Triangles Recall from Chapter 5, we studied various ways to prove triangles to be congruent: In §5-6, we studied two theorems R S T A B C ASA and R S T A B C AAS

SAS The theorems mentioned in Chapter 5, were for ALL triangles.
Right Triangles The theorems mentioned in Chapter 5, were for ALL triangles. So, it should make perfect sense that they would apply to right triangles as well. Theorem 6-6 LL Theorem If two legs of one right triangle are congruent to the corresponding legs of another right triangle, then the triangles are congruent. A C B D F E SAS same as

AAS The theorems mentioned in Chapter 5, were for ALL triangles.
Right Triangles The theorems mentioned in Chapter 5, were for ALL triangles. So, it should make perfect sense that they would apply to right triangles as well. Theorem 6-7 HA Theorem If ______________ and an (either) __________ of one right triangle are congruent to the __________ and _________________ of another right angle, then the triangles are congruent. the hypotenuse acute angle hypotenuse corresponding angle A C B D F E AAS same as

ASA The theorems mentioned in Chapter 5, were for ALL triangles.
Right Triangles The theorems mentioned in Chapter 5, were for ALL triangles. So, it should make perfect sense that they would apply to right triangles as well. Theorem 6-6 LA Theorem If one (either) ___ and an __________ of a right triangle are congruent to the ________________________ of another right triangle, then the triangles are congruent. leg acute angle corresponding leg and angle A C B D F E ASA same as

ASS The theorems mentioned in Chapter 5, were for ALL triangles.
Right Triangles The theorems mentioned in Chapter 5, were for ALL triangles. So, it should make perfect sense that they would apply to right triangles as well. Postulate 6-1 HL Postulate If the hypotenuse and a leg on one right triangle are congruent to the hypotenuse and corresponding leg of another right triangle, then the triangles are congruent. A C B D F E ASS Theorem?

End of Section 6.5

Pythagorean Theorem What You'll Learn You will learn to use the __________ Theorem and its converse. Pythagorean 1) _________________ 2) _______________ * 3) _______ Vocabulary Pythagorean Theorem Pythagorean triple converse

If ___ measures of the sides of a _____ triangle are known, the
Pythagorean Theorem If ___ measures of the sides of a _____ triangle are known, the ___________________ can be used to find the measure of the third ____. two right Pythagorean Theorem side c a b A _________________ is a group of three whole numbers that satisfies the equation c2 = a2 + b2, where c is the measure of the hypotenuse. Pythagorean triple 3 5 4 52 = 25 =

lengths of the legs __ and __.
Pythagorean Theorem Theorem 6-9 Pythagorean Theorem In a right triangle, the square of the length of the hypotenuse __, is equal to the sum of the squares of the lengths of the legs __ and __. c a b c a b Theorem 6-10 Converse of the Pythagorean Theorem If c is the measure of the longest side of a triangle, a and b are the lengths of the other two sides, and c2 = a2 + b2, then the triangle is a right angle.

End of Section 6.6

Distance on the Coordinate Plane
What You'll Learn You will learn to find the ______________________on the coordinate plane. distance between two points Nothing new! You learned this in Algebra I. Vocabulary

Hands-On! y x C(2, 3) 1) On grid paper, graph A(-3, 1) and C(2, 3). A(-3, 1) 2) Draw a horizontal segment from A and a vertical segment from C. B(2, 1) 3) Label the intersection B and find the coordinates of B. QUESTIONS: What is the measure of the distance between A and B? (x2 – x1) = 5 What is the measure of the distance between B and C? (y2 – y1) = 2 What kind of triangle is ΔABC? right triangle If AB and BC are known, what theorem can be used to find AC? Pythagorean Theorem What is the measure of AC? 29 ≈ 5.4

Theorem 6-11 Distance Formula If d is the measure of the distance between two points with coordinates (x1, y1) and (x2, y2), y x A(x1, y1) B(x2, y2) d then d =

Find the distance between each pair of points
Find the distance between each pair of points. Round to the nearest tenth, if necessary. 5 4.5

End of Section 6.7

Triangle Inequalities
§ 7.1 Segments, Angles, and Inequalities § 7.2 Exterior Angle Theorem § 7.3 Inequalities Within a Triangle § 7.4 Triangle Inequality Theorem 322

Segments, Angles, and Inequalities
What You'll Learn You will learn to apply inequalities to segment and angle measures. Inequalities Vocabulary 1) Inequality 323

The Comparison Property of Numbers is used to compare two line segments of unequal measures. The property states that given two unequal numbers a and b, either: a < b or a > b T U 2 cm V W 4 cm The length of is less than the length of , or TU < VW The same property is also used to compare angles of unequal measures. 324

Segments, Angles, and Inequalities Postulate 7 – 1 Comparison Property
J 133° K 60° The measure of J is greater than the measure of K. The statements TU > VW and J > K are called __________ because they contain the symbol < or >. inequalities Postulate 7 – 1 Comparison Property For any two real numbers, a and b, exactly one of the following statements is true. a < b a = b a > b 325

6 4 2 -2 Replace with <, >, or = to make a true statement. SN > DN Lesson 2-1 Finding Distance on a number line. 6 – (- 1) 6 – 2 7 > 4 326

Theorem 7 – 1 If point C is between points A and B, and A, C, and B are collinear, then ________ and ________. AB > AC AB > CB A C B A similar theorem for comparing angle measures is stated below. This theorem is based on the Angle Addition Postulate. 327

A similar theorem for comparing angle measures is stated below. This theorem is based on the Angle Addition Postulate. Theorem 7 – 2 D P F E 328

Use theorem 7 – 2 to solve the following problem. Replace with <, >, or = to make a true statement. 108° 149° 45° 40° 18° A B C D mBDA < mCDA Check: 45° 40° + 45° 45° < 85° 329

Property Transitive Property For any numbers a, b, and c, 1) if a < b and b < c, then a < c. if 5 < 8 and 8 < 9, then 5 < 9. 2) if a > b and b > c, then a > c. if 7 > 6 and 6 > 3, then 7 > 3. 330

Property Addition and Subtraction Properties Multiplication and Division Properties For any numbers a, b, and c, 1) if a < b, then a + c < b + c and a – c < b – c. 1 < 3 1 + 5 < 3 + 5 6 < 8 2) if a > b, then a + c > b + c and a – c > b – c. For any numbers a, b, and c, 331

End of Section 7.1

Exterior Angle Theorem
What You'll Learn You will learn to identify exterior angles and remote interior angles of a triangle and use the Exterior Angle Theorem. Vocabulary 1) Interior angle 2) Exterior angle 3) Remote interior angle 333

interior In the triangle below, recall that 1, 2, and 3 are _______ angles of ΔPQR. Angle 4 is called an _______ angle of ΔPQR. exterior An exterior angle of a triangle is an angle that forms a _________ with one of the angles of the triangle. linear pair In ΔPQR, 4 is an exterior angle at R because it forms a linear pair with 3. ____________________ of a triangle are the two angles that do not form a linear pair with the exterior angle. Remote interior angles In ΔPQR, 1, and 2 are the remote interior angles with respect to 4. 1 2 3 4 P Q R 334

In the figure below, 2 and 3 are remote interior angles with respect to what angle? 5 1 2 3 4 5 335

The measure of an exterior angle of a triangle is equal to sum of the measures of its ___________________. remote interior angles X 4 3 2 1 Z Y m4 = m1 + m2 336

Inequality Theorem The measure of an exterior angle of a triangle is greater than the measures of either of its two ____________________. remote interior angles X 4 3 2 1 Z Y m4 > m1 m4 > m2 338

Name two angles in the triangle below that have measures less than 74°. 1 and 3 74° 1 3 2 Theorem 7 – 5 If a triangle has one right angle, then the other two angles must be _____. acute 339

The feather–shaped leaf is called a pinnatifid. In the figure, does x = y? Explain. 28° x = y ? __ = 28 109 = 110 No! x does not equal y

End of Section 7.2

Inequalities Within a Triangle
What You'll Learn You will learn to identify the relationships between the _____ and _____ of a triangle. sides angles Vocabulary Nothing New! 343

Theorem 7 – 6 If the measures of three sides of a triangle are unequal, then the measures of the angles opposite those sides are unequal ________________. in the same order 13 8 11 L P M LP < PM < ML mM < mL < mP 344

Theorem 7 – 7 If the measures of three angles of a triangle are unequal, then the measures of the sides opposite those angles are unequal ________________. in the same order J 45° W K 60° 75° mW < mJ < mK JK < KW < WJ 345

Theorem 7 – 8 In a right triangle, the hypotenuse is the side with the ________________. greatest measure Y W X 5 3 4 WY > XW WY > XY 346

The longest side is The largest angle is So, the largest angle is So, the longest side is 347

End of Section 7.3

Triangle Inequality Theorem
What You'll Learn You will learn to identify and use the Triangle Inequality Theorem. Vocabulary Nothing New! 349

The sum of the measures of any two sides of a triangle is _______ than the measure of the third side. greater a b c a + b > c a + c > b b + c > a 350

Can 16, 10, and 5 be the measures of the sides of a triangle? No! > 5 > 10 However, > 16 351

End of Section 7.4

Quadrilaterals § 8.1 Quadrilaterals § 8.2 Parallelograms
§ 8.3 Tests for Parallelograms § 8.4 Rectangles, Rhombi, and Squares § 8.5 Trapezoids

What You'll Learn Vocabulary Quadrilaterals
You will learn to identify parts of quadrilaterals and find the sum of the measures of the interior angles of a quadrilateral. Vocabulary 1) Quadrilateral 2) Consecutive 3) Nonconsecutive 4) Diagonal 354

The segments of a quadrilateral intersect only at their endpoints.
Quadrilaterals A quadrilateral is a closed geometric figure with ____ sides and ____ vertices. four four Quadrilaterals Not Quadrilaterals The segments of a quadrilateral intersect only at their endpoints. Special types of quadrilaterals include squares and rectangles. 355

Quadrilaterals are named by listing their vertices in order.
There are several names for the quadrilateral below. Some examples: quadrilateral ABCD B quadrilateral BCDA A quadrilateral CDAB or quadrilateral DABC D C 356

Quadrilaterals Any two _______ of a quadrilateral are either __________ or _____________. vertices angles sides consecutive nonconsecutive S R Q P 357

R and P are nonconsecutive vertices.
Quadrilaterals Segments that join nonconsecutive vertices of a quadrilateral are called ________. diagonals S R Q P R and P are nonconsecutive vertices. S and Q are nonconsecutive vertices. 358

Name all pairs of consecutive sides:
Quadrilaterals Name all pairs of consecutive sides: Q T S R Name all pairs of nonconsecutive angles: Name the diagonals: 359

Considering the quadrilateral to the right.
Quadrilaterals D C B A Considering the quadrilateral to the right. 1 What shapes are formed if a diagonal is drawn? ___________ two triangles 2 3 5 4 Use the Angle Sum Theorem (Section 5-2) to find m1 + m2 + m3 180 Use the Angle Sum Theorem (Section 5-2) to find m4 + m5 + m6 180 6 Find m1 + m2 + m m4 + m5 + m6 180 + 180 360 This leads to the following theorem. 360

The sum of the measures of the angles of a quadrilateral is ____.
Quadrilaterals Theorem 8-1 The sum of the measures of the angles of a quadrilateral is ____. 360 a° d° c° b° a + b + c + d = 360 361

Quadrilaterals Find the measure of B in quadrilateral ABCD if A = x, B = 2x, C = x – 10, and D = 50. mA + mB + mC + mD = 360 A D C B x + 2x + x – = 360 4x = 360 4x = 320 x = 80 B = 2x B = 2(80) B = 160 362

Quadrilaterals End of Section 8.1 363

What You'll Learn Vocabulary Parallelograms
You will learn to identify and use the properties of parallelograms. Vocabulary 1) Parallelogram 364

A parallelogram is a quadrilateral with two pairs of ____________.
Parallelograms A parallelogram is a quadrilateral with two pairs of ____________. parallel sides In parallelogram ABCD below, and A B D C Also, the parallel sides are _________. congruent Knowledge gained about “parallels” (chapter 4) will now be used in the following theorems. 365

Opposite angles of a parallelogram are ________.
Parallelograms Theorem 8-2 Theorem 8-3 Theorem 8-4 A B D C Opposite angles of a parallelogram are ________. congruent A  C and B  D A B D C Opposite sides of a parallelogram are ________. congruent The consecutive angles of a parallelogram are ____________. A B D C supplementary mA + mB = 180 mD + mC = 180 366

Parallelograms In RSTU, RS = 45, ST = 70, and U = 68. Find: RU = ____
68° Find: RU = ____ 70 Theorem 8-3 UT = _____ 45 Theorem 8-3 mS = _____ 68° Theorem 8-2 mT = _____ 112° Theorem 8-4 367

The diagonals of a parallelogram ______ each other.
Parallelograms Theorem 8-5 The diagonals of a parallelogram ______ each other. bisect A D B C E In RSTU, if RT = 56, find RE. R U S T E RE = 28 368

In the figure below, ABCD is a parallelogram.
Parallelograms In the figure below, ABCD is a parallelogram. Since AD || BC and diagonal DB is a transversal, then ADB  CBD. A D B C (Alternate Interior angles) Since AB || DC and diagonal DB is a transversal, then BDC  DBA. (Alternate Interior angles) DB  BD ASA Theorem 369

A diagonal of a parallelogram separates it into two _________________.
Theorem 8-6 A diagonal of a parallelogram separates it into two _________________. congruent triangles A D B C 370

Parallelograms parallelogram
The Escher design below is based on a _____________. parallelogram You can use a parallelogram to make a simple Escher-like drawing. Change one side of the parallelogram and then translate (slide) the change to the opposite side. The resulting figure is used to make a design with different colors and textures. 371

Parallelograms End of Section 8.2 372

Tests for Parallelograms
What You'll Learn You will learn to identify and use tests to show that a quadrilateral is a parallelogram. Vocabulary Nothing New! 373

Theorem 8-7 If both pairs of opposite sides of a quadrilateral are _________, then the quadrilateral is a parallelogram. congruent A D C B 374

You can use the properties of congruent triangles and Theorem 8-7 to find other ways to show that a quadrilateral is a parallelogram. In quadrilateral PQRS, PR and QS bisect each other at T. T P S R Q Show that PQRS is a parallelogram by providing a reason for each step. Definition of segment bisector Vertical angles are congruent SAS Corresp. parts of Congruent Triangles are Congruent Theorem 8-7 375

Theorem 8-8 If one pair of opposite sides of a quadrilateral is _______ and _________, then the quadrilateral is a parallelogram. parallel congruent A D C B 376

Theorem 8-9 If the diagonals of a quadrilateral ________________, then the quadrilateral is a parallelogram. bisect each other A D C B E 377

Determine whether each quadrilateral is a parallelogram. If the figure is a parallelogram, give a reason for your answer. A D C B Given Alt. Int. Angles Therefore, quadrilateral ABCD is a parallelogram. Theorem 8-8 378

End of Section 8.3 379

Rectangles, Rhombi, and Squares
What You'll Learn You will learn to identify and use the properties of rectangles, rhombi, and squares. Vocabulary 1) Rectangle 2) Rhombus 3) Square 380

A closed figure, 4 sides & 4 vertices Quadrilateral Opposite sides parallel opposite sides congruent Parallelogram Parallelogram with 4 congruent sides Parallelogram with 4 right angles Rhombus Rectangle Parallelogram with 4 congruent sides and 4 right angles Square 381

Identify the parallelogram below. Identify the parallelogram below. D C B A rhombus Parallelogram ABCD has 4 right angles, but the four sides are not congruent. Therefore, it is a _________ rectangle 382

Theorem 8-10 The diagonals of a rectangle are _________. congruent A D B C 383

Theorem 8-11 The diagonals of a rhombus are ____________. perpendicular A B D C 384

Theorem 8-12 Each diagonal of a rhombus _______ a pair of opposite angles. bisects A 1 2 D 8 7 6 5 C 3 4 B 385

Use square XYZW to answer the following questions: 1) If YW = 14, XZ = ____ 14 W Z Y X O A square has all the properties of a rectangle, and the diagonals of a rectangle are congruent. 2) mYOX = ____ 90 A square has all the properties of a rhombus, and the diagonals of a rhombus are perpendicular. 3) Name all segments that are congruent to WO. Explain your reasoning. The diagonals are congruent and they bisect each other. OY, XO, and OZ 386

Use the Venn diagram to answer the following questions: T or F Quadrilaterals Parallelograms Rhombi Rectangles Squares 1) Every square is a rhombus: ___ T 5) All rhombi are parallelograms: ___ T 2) Every rhombus is a square: ___ F 3) Every rectangle is a square: ___ F 6) Every parallelogram is a rectangle: ___ F 4) Every square is a rectangle: ___ T 387

What You'll Learn Vocabulary Trapezoids
You will learn to identify and use the properties of trapezoids and isosceles trapezoids. Vocabulary 1) Trapezoid 389

A trapezoid is a ____________ with exactly one pair of ____________.
Trapezoids A trapezoid is a ____________ with exactly one pair of ____________. quadrilateral parallel sides The parallel sides are called ______. bases The non parallel sides are called _____. legs base T P A R Each trapezoid has two pair of base angles. leg leg base angles T and R are one pair of base angles. P and A are the other pair of base angles. base 390

The median of a trapezoid is parallel to the _____, bases
Trapezoids Theorem 8-13 The median of a trapezoid is parallel to the _____, bases and the length of the median equals _______________ of the lengths of the bases. one-half the sum C N B D M A 391

Find the length of median MN in trapezoid ABCD if AB = 16 and DC = 20
Trapezoids Find the length of median MN in trapezoid ABCD if AB = 16 and DC = 20 16 C N D M B A 18 20 392

If the legs of a trapezoid are congruent, the trapezoid is an
Trapezoids If the legs of a trapezoid are congruent, the trapezoid is an _________________. isosceles trapezoid In lesson 6 – 4, you learned that the base angles of an isosceles triangle are congruent. There is a similar property of isosceles trapezoids. 393

Each pair of __________ in an isosceles trapezoid is congruent.
Trapezoids Theorem 8-14 Each pair of __________ in an isosceles trapezoid is congruent. base angles Z Y X W 394

Find the missing angle measures in isosceles trapezoid TRAP.
Trapezoids Find the missing angle measures in isosceles trapezoid TRAP. T P A R 60° P  A Theorem 8 – 14 120° 120° mP = mA 60 = mA T  R Theorem 8 – 14 60° P + A + 2(T) = 360 (T) = 360 (T) = 360 2(T) = 240 T = 120 R = 120 395

Trapezoids End of Section 8.5 396

Proportions and Similarity
§ 9.1 Using Ratios and Proportions § 9.2 Similar Polygons § 9.3 Similar Triangles § 9.4 Proportional Parts and Triangles § 9.5 Triangles and Parallel Lines § 9.6 Proportional Parts and Parallel Lines § 9.7 Perimeters and Similarity 397

Using Ratios and Proportions
What You'll Learn You will learn to use ratios and proportions to solve problems. Vocabulary 1) ratio 2) proportion 3) cross products 4) extremes 5) means 398

In 2000, about 180 million tons of solid waste was created in the United States. The paper made up about 72 million tons of this waste. The ratio of paper waste to total waste is 72 to 180. This ratio can be written in the following ways. 72 to 180 72:180 72 ÷ 180 Definition of Ratio A ratio is a comparison of two numbers by division. a to b a:b a ÷ b where b  0 399

A __________ is an equation that shows two equivalent ratios. proportion Every proportion has two cross products. In the proportion to the right, the terms 20 and 3 are called the extremes, and the terms 30 and 2 are called the means. 30(2) = 20(3) The cross products are 20(3) and 30(2). 60 = 60 The cross products are always _____ in a proportion. equal 400

Theorem 9-1 Property of Proportions For any numbers a and c and any nonzero numbers b and d, Likewise, 401

Solve each proportion: 15(2x) = 30(6) 3(x) = (30 – x)2 30x = 180 3x = 60 – 2x x = 6 5x = 60 x = 12 402

Driving gear Driven gear The gear ratio is the number of teeth on the driving gear to the number of teeth on the driven gear. If the gear ratio is 5:2 and the driving gear has 35 teeth, how many teeth does the driven gear have? given ratio equivalent ratio = driving gear driven gear 5 35 driving gear = 2 x driven gear 5x = 70 x = 14 The driven gear has 14 teeth. 403

Similar Polygons What You'll Learn You will learn to identify similar polygons. Vocabulary 1) polygons 2) sides 3) similar polygons 4) scale drawing 405

Polygons that are the same shape but not necessarily the same size are
Similar Polygons A polygon is a ______ figure in a plane formed by segments called sides. closed It is a general term used to describe a geometric figure with at least three sides. Polygons that are the same shape but not necessarily the same size are called ______________. similar polygons The symbol ~ is used to show that two figures are similar. ΔABC is similar to ΔDEF A B C D F E ΔABC ~ ΔDEF 406

Polygon ABCD ~ polygon EFGH
Similar Polygons Definition of Similar Polygons Two polygons are similar if and only if their corresponding angles are congruent and the measures of their corresponding sides are ___________. proportional C D A B F E G H and Polygon ABCD ~ polygon EFGH 407

Determine if the polygons are similar. Justify your answer.
Similar Polygons Determine if the polygons are similar. Justify your answer. 7 6 6 4 5 4 5 7 1) Are corresponding angles are _________. congruent 2) Are corresponding sides ___________. proportional = 0.66 = 0.71 The polygons are NOT similar! 408

Find the values of x and y if ΔRST ~ ΔJKL
Similar Polygons Find the values of x and y if ΔRST ~ ΔJKL R T S J L K 4 5 6 7 x y + 2 Write the proportion that can be solved for y. 4 6 = y + 2 7 Write the proportion that can be solved for x. 4(y + 2) = 42 4y + 8 = 42 4 5 = 4y = 34 7 x 4x = 35 409

Contractors use scale drawings to represent the floorplan of a house.
Similar Polygons Scale drawings are often used to represent something that is too large or too small to be drawn at actual size. Contractors use scale drawings to represent the floorplan of a house. Dining Room Kitchen Living Garage Utility 1.25 in. .75 in. 1 in. .5 in. Scale: 1 in. = 16 ft. Use proportions to find the actual dimensions of the kitchen. width length 1 in 1.25 in. 1 in .75 in. = = 16 ft w ft. 16 ft L ft. (16)(1.25) = w (16)(.75) = L 20 = w 12 = L width is 20 ft. length is 12 ft. 410

Similar Polygons End of Section 9.2 411

Similar Triangles What You'll Learn You will learn to use AA, SSS, and SAS similarity tests for triangles. Vocabulary Nothing New! 412

Some of the triangles are similar, as shown below.
Similar Triangles Some of the triangles are similar, as shown below. The Bank of China building in Hong Kong is one of the ten tallest buildings in the world. Designed by American architect I.M. Pei, the outside of the 70-story building is sectioned into triangles which are meant to resemble the trunk of a bamboo plant. 413

three corresponding parts of each triangle.
Similar Triangles In previous lessons, you learned several basic tests for determining whether two triangles are congruent. Recall that each congruence test involves only three corresponding parts of each triangle. Likewise, there are tests for similarity that will not involve all the parts of each triangle. Postulate 9-1 AA Similarity If two angles of one triangle are congruent to two corresponding angles of another triangle, then the triangles are ______. similar C F A D E B If A ≈ D and B ≈ E, then ΔABC ~ ΔDEF 414

If the measures of the sides of a triangle are ___________
Similar Triangles Two other tests are used to determine whether two triangles are similar. Theorem 9-2 SSS Similarity If the measures of the sides of a triangle are ___________ to the measures of the corresponding sides of another triangle, then the triangles are similar. proportional C 6 F 2 3 1 A B D E 8 4 then the triangles are similar then ΔABC ~ ΔDEF 415

If the measures of two sides of a triangle are ___________
Similar Triangles Theorem 9-3 SAS Similarity If the measures of two sides of a triangle are ___________ to the measures of two corresponding sides of another triangle and their included angles are congruent, then the triangles are similar. proportional C F 2 1 A B D 4 E 8 then ΔABC ~ ΔDEF 416

is used and complete the statement.
Similar Triangles Determine whether the triangles are similar. If so, tell which similarity test is used and complete the statement. G H K M P J 6 14 10 9 15 21 6 10 14 Since , the triangles are similar by SSS similarity. = = 9 15 21 Therefore, ΔGHK ~ Δ JMP 417

If Fransisco is 6 feet tall, how tall is the tree?
Similar Triangles Fransisco needs to know the tree’s height. The tree’s shadow is 18 feet long at the same time that his shadow is 4 feet long. If Fransisco is 6 feet tall, how tall is the tree? 1) The sun’s rays form congruent angles with the ground. 2) Both Fransisco and the tree form right angles with the ground. 4 6 = t 18 4t = 108 t = 27 6 ft. The tree is 27 feet tall! 4 ft. 18 ft. 418

Similar Triangles Slade is a surveyor. To find the distance across Muddy Pond, he forms similar triangles and measures distances as shown. 8 m 10 m 45 m x What is the distance across Muddy Pond? 10 8 = It is 36 meters across Muddy Pond! 45 x 10x = 360 x = 36 419

Similar Triangles End of Section 9.3 420

Proportional Parts and Triangles
What You'll Learn You will learn to identify and use the relationships between proportional parts of triangles. Vocabulary Nothing New! 421

In ΔPQR, and intersects the other two sides of ΔPQR. Are ΔPQR and ΔPST, similar? R Q P PST  PQR corresponding angles P  P ΔPQR ~ ΔPST. Why? (What theorem / postulate?) S T AA Similarity (Postulate 9-1) 422

Theorem 9-4 If a line is _______ to one side of a triangle, and intersects the other two sides, then the triangle formed is _______ to the original triangle. parallel similar A B C D E If ΔABC ~ ΔADE. 423

Since , ΔSVW ~ ΔSRT. Complete the proportion: T W S V R SV 424

Theorem 9-5 If a line is parallel to one side of a triangle and intersects the other two sides, then it separates the sides into segments of __________________. proportional lengths E D C B A 425

C B H G A 5 3 x + 5 x 4 426

Jacob is a carpenter. Needing to reinforce this roof rafter, he must find the length of the brace. 10 ft 6 ft 4 ft Brace 4 x 4 ft x = 10 4 10x = 16 x = 1 3 5 ft 427

Triangles and Parallel Lines
What You'll Learn You will learn to use proportions to determine whether lines are parallel to sides of triangles. Vocabulary Nothing New! 429

You know that if a line is parallel to one side of a triangle and intersects the other two sides, then it separates the sides into segments of proportional lengths (Theorem 9-5). The converse of this theorem is also true. Theorem 9-6 If a line intersects two sides of a triangle and separates the sides into corresponding segments of proportional lengths, then the line is parallel to the third side. E D C B A 9 6 4 430

Theorem 9-7 If a segment joins the midpoints of two sides of a triangle, then it is parallel to the third side, and its measure equals ________ the measure of the third side. one-half E D C B A 2x x 431

Use theorem 9 – 7 to find the length of segment DE. A 5 8 11 x D E 22 B C 432

A, B, and C are midpoints of the sides of ΔMNP. C B A N P M Complete each statement. 1) MP || ____ AC 2) If BC = 14, then MN = ____ 28 3) If mMNP = s, then mBCP = ___ s 4) If MP = 18x, then AC = __ 9x 433

A, B, and C are midpoints of the sides of ΔDEF. C B A D F E 8 7 5 1) Find DE, EF, and FD. 14; 10; 16 2) Find the perimeter of ΔABC 20 3) Find the perimeter of ΔDEF 40 4) Find the ratio of the perimeter of ΔABC to the perimeter of ΔDEF. 20:40 = 1:2 434

ABCD is a quadrilateral. A D C B E is the midpoint of AD F F is the midpoint of DC E H H is the midpoint of CB G is the midpoint of BA G Q1) What can you say about EF and GH ? They are parallel (Hint: Draw diagonal AC .) Q2) What kind of figure is EFHG ? Parallelogram 435

Proportional Parts and Parallel Lines
What You'll Learn You will learn to identify and use the relationships between parallel lines and proportional parts. Vocabulary Nothing New! 437

Hands-On On your given paper, draw two (transversals) lines intersecting the parallel lines. A D Label the intersections of the transversals and the parallel lines, as shown here. B E F C Measure AB, BC, DE, and EF. , Calculate each set of ratios: BC AB EF DE AC AB DF DE , Do the parallel lines divide the transversals proportionally? Yes 438

Theorem 9-8 If three or more parallel lines intersect two transversals, the lines divide the transversals proportionally. l m n B A C F E D If l || m || n Then BC AB EF DE = , AC AB DF DE = and , AC BC DF EF = 439

Find the value of x. a b c H G J W V U 18 12 x 15 GH UV = HJ VW 12 15 = 18 x 12x = 18(15) 12x = 270 x = 22 2 1 440

Theorem 9-9 If three or more parallel lines cut off congruent segments on one transversal, then they cut off congruent segments on every transversal. l m n B A C F E D If l || m || n and AB  BC, Then DE  EF. 441

Find the value of x. A 10 Since AB  BC, B 10 DE  EF Theorem 9 - 9 C (x + 3) = (2x – 2) x + 3 = 2x – 2 5 = x (2x – 2) (x +3) 8 8 F D E 442

Perimeters and Similarity
What You'll Learn You will learn to identify and use proportional relationships of similar triangles. Vocabulary 1) Scale Factor 444

These right triangles are similar! Therefore, the measures of their corresponding sides are ___________. proportional Use the ____________ theorem to calculate the length of the hypotenuse. Pythagorean 10 6 15 9 8 12 6 8 10 2 We know that = = = 9 12 15 3 Is there a relationship between the measures of the perimeters of the two triangles? perimeter of small Δ perimeter of large Δ = = 36 24 = 3 2 445

Theorem 9-10 If two triangles are similar, then the measures of the corresponding perimeters are proportional to the measures of the corresponding sides. A C B F E D If ΔABC ~ ΔDEF, then perimeter of ΔABC perimeter of ΔDEF = DE AB = EF BC = FD CA 446

The perimeter of ΔRST is 9 units, and ΔRST ~ ΔMNP. Find the value of each variable. M N 4.5 P R S T 3 6 z Y x perimeter of ΔMNP perimeter of ΔRST RS MN = RS MN = ST NP RS MN = TR PM Theorem 9-10 2 3 = y 6 2 3 = z 4.5 13.5 9 x 3 = The perimeter of ΔMNP is 3y = 12 3z = 9 27 = 13.5x Cross Products x = 2 y = 4 z = 3 447

The ratio found by comparing the measures of corresponding sides of similar triangles is called the constant of proportionality or the ___________. scale factor D E F 14 10 6 A B C 7 5 3 DE AB = EF BC FD CA If ΔABC ~ ΔDEF, then or 6 3 = 14 7 10 5 2 1 The scale factor of ΔABC to ΔDEF is 2 1 Each ratio is equivalent to 1 2 The scale factor of ΔDEF to ΔABC is 448

Hands-On A Step 1) On a piece of lined paper, pick a point on one of the lines and label it A. Use a straightedge and protractor to draw A so that mA < 90 and only the vertex lies on the line. Step 2) Extend one side of A down four lines. Label this point E. Do the same for the other side of A. Label this point I. Now connect points E and I to form ΔAEI. Step 3) Label the points where the horizontal lines intersect segment AG (B through D). Label the points where the horizontal lines intersect segment AI (F through H). B F C G D H E I What can you conclude about the lines through the sides of ΔAEI and parallel to segment EI? This activity suggests Theorem 9-5. 450

Polygons and Area § 10.1 Naming Polygons
§ Diagonals and Angle Measure § Areas of Polygons § Areas of Triangles and Trapezoids § Areas of Regular Polygons § Symmetry § Tessellations 451

Naming Polygons What You'll Learn You will learn to name polygons according to the number of _____ and ______. sides angles Vocabulary 1) regular polygon 2) convex 3) concave 452

A polygon is named by the number of its _____ or ______. sides angles
Naming Polygons A polygon is a _____________ in a plane formed by segments, called sides. closed figure A polygon is named by the number of its _____ or ______. sides angles A triangle is a polygon with three sides. The prefix ___ means three. tri

Prefixes are also used to name other polygons. Prefix Number of Sides
Naming Polygons Prefixes are also used to name other polygons. Prefix Number of Sides Name of Polygon tri- 3 triangle quadri- 4 quadrilateral penta- 5 pentagon hexa- 6 hexagon hepta- 7 heptagon octa- 8 octagon nona- 9 nonagon deca- 10 decagon

Terms Naming Polygons A vertex is the point of intersection of
two sides. Consecutive vertices are the two endpoints of any side. U T S Q R P A segment whose endpoints are nonconsecutive vertices is a diagonal. Sides that share a vertex are called consecutive sides.

An equilateral polygon has all _____ congruent. sides
Naming Polygons An equilateral polygon has all _____ congruent. sides An equiangular polygon has all ______ congruent. angles A regular polygon is both ___________ and ___________. equilateral equiangular equilateral but not equiangular equiangular but not equilateral regular, both equilateral and equiangular Investigation: As the number of sides of a series of regular polygons increases, what do you notice about the shape of the polygons?

A polygon can also be classified as convex or concave.
Naming Polygons A polygon can also be classified as convex or concave. If all of the diagonals lie in the interior of the figure, then the polygon is ______. If any part of a diagonal lies outside of the figure, then the polygon is _______. concave convex

Naming Polygons End of Section 10.1

Diagonals and Angle Measure
What You'll Learn You will learn to find measures of interior and exterior angles of polygons. Vocabulary Nothing New!

Diagonals and Angle Measure Number of Diagonals from One Vertex
Make a table like the one below. 1) Draw a convex quadrilateral. 2) Choose one vertex and draw all possible diagonals from that vertex. 3) How many triangles are formed? Convex Polygon Number of Sides Number of Diagonals from One Vertex Number of Triangles Sum of Interior Angles quadrilateral 4 1 2 2(180) = 360

1) Draw a convex pentagon. 2) Choose one vertex and draw all possible diagonals from that vertex. 3) How many triangles are formed? Convex Polygon Number of Sides Number of Diagonals from One Vertex Number of Triangles Sum of Interior Angles quadrilateral 4 1 2 2(180) = 360 pentagon 5 2 3 3(180) = 540

1) Draw a convex hexagon. 2) Choose one vertex and draw all possible diagonals from that vertex. 3) How many triangles are formed? Convex Polygon Number of Sides Number of Diagonals from One Vertex Number of Triangles Sum of Interior Angles quadrilateral 4 1 2 2(180) = 360 pentagon 5 2 3 3(180) = 540 hexagon 6 3 4 4(180) = 720

1) Draw a convex heptagon. 2) Choose one vertex and draw all possible diagonals from that vertex. 3) How many triangles are formed? Convex Polygon Number of Sides Number of Diagonals from One Vertex Number of Triangles Sum of Interior Angles quadrilateral 4 1 2 2(180) = 360 pentagon 5 2 3 3(180) = 540 hexagon 6 3 4 4(180) = 720 heptagon 7 4 5 5(180) = 900

1) Any convex polygon. 2) All possible diagonals from one vertex. 3) How many triangles? Convex Polygon Number of Sides Number of Diagonals from One Vertex Number of Triangles Sum of Interior Angles quadrilateral 4 1 2 2(180) = 360 pentagon 5 2 3 3(180) = 540 hexagon 6 3 4 4(180) = 720 heptagon 7 4 5 5(180) = 900 n-gon n n - 3 n - 2 (n – 2)180 Theorem 10-1 If a convex polygon has n sides, then the sum of the measure of its interior angles is (n – 2)180.

In §7.2 we identified exterior angles of triangles. Likewise, you can extend the sides of any convex polygon to form exterior angles. 48° 57° 74° The figure suggests a method for finding the sum of the measures of the exterior angles of a convex polygon. 72° 55° 54° When you extend n sides of a polygon, n linear pairs of angles are formed. The sum of the angle measures in each linear pair is 180. sum of measure of exterior angles sum of measures of linear pairs sum of measures of interior angles = – = n•180 – 180(n – 2) = 180n – 180n + 360 sum of measure of exterior angles = 360

Theorem 10-2 In any convex polygon, the sum of the measures of the exterior angles, (one at each vertex), is 360. Java Applet

End of Section 10.2

Areas of Polygons What You'll Learn You will learn to calculate and estimate the areas of polygons. Vocabulary 1) polygonal region 2) composite figure 3) irregular figure

Any polygon and its interior are called a ______________.
Areas of Polygons Any polygon and its interior are called a ______________. polygonal region In lesson 1-6, you found the areas of rectangles. Postulate 10-1 Area Postulate For any polygon and a given unit of measure, there is a unique number A called the measure of the area of the polygon. Area can be used to describe, compare, and contrast polygons. The two polygons below are congruent. How do the areas of these polygons compare? They are the same. Postulate 10-2 Congruent polygons have equal areas.

The figures above are examples of ________________. composite figures
Areas of Polygons The figures above are examples of ________________. composite figures They are each made from a rectangle and a triangle that have been placed together. You can use what you know about the pieces to gain information about the figure made from them. You can find the area of any polygon by dividing the original region into smaller and simpler polygon regions, like _______, __________, and ________. squares rectangles triangles The area of the original polygonal region can then be found by __________ _________________________. adding the areas of the smaller polygons

Area Addition Postulate
Areas of Polygons Postulate 10-3 Area Addition Postulate The area of a given polygon equals the sum of the areas of the non-overlapping polygons that form the given polygon. 1 2 3 AreaTotal = A1 + A2 + A3

Find the area of the polygon in square units.
Areas of Polygons Find the area of the polygon in square units. Area of Square 3u X 3u = 9u2 Area of Rectangle 1u X 2u = 2u2 Area of polygon = = 7u2 3 units Area of Rectangle Area of Square

Areas of Polygons End of Section 10.3

Areas of Triangles and Trapezoids
What You'll Learn You will learn to find the areas of triangles and trapezoids. Vocabulary Nothing new! 474

Look at the rectangle below. Its area is bh square units. The diagonal divides the rectangle into two _________________. congruent triangles The area of each triangle is half the area of the rectangle, or This result is true of all triangles and is formally stated in Theorem 10-3. b h 475

Height Consider the area of this rectangle A(rectangle) = bh Base 476

Theorem 10-3 Area of a Triangle If a triangle has an area of A square units, a base of b units, and a corresponding altitude of h units, then h b 477

Find the area of each triangle: 6 yd 18 mi 23 mi A = 13 yd2 A = 207 mi2 478

Because the opposite sides of a parallelogram have the same length, the area of a parallelogram is closely related to the area of a ________. Next we will look at the area of trapezoids. However, it is helpful to first understand parallelograms. rectangle height base The area of a parallelogram is found by multiplying the ____ and the ______. base height Base – the bottom of a geometric figure. Height – measured from top to bottom, perpendicular to the base. 479

Starting with a single trapezoid. The height is labeled h, and the bases are labeled b1 and b2 Construct a congruent trapezoid and arrange it so that a pair of congruent legs are adjacent. b1 b2 b1 h b2 The new, composite figure is a parallelogram. It’s base is (b1 + b2) and it’s height is the same as the original trapezoid. The area of the parallelogram is calculated by multiplying the base X height. A(parallelogram) = h(b1 + b2) The area of the trapezoid is one-half of the parallelogram’s area. 480

Theorem 10-4 Area of a Trapezoid If a trapezoid has an area of A square units, bases of b1 and b2 units, and an altitude of h units, then b1 b2 h 481

Find the area of the trapezoid: 18 in 20 in 38 in A = 522 in2 482

End of Lesson 483

Areas of Regular Polygons
What You'll Learn You will learn to find the areas of regular polygons. Vocabulary 1) center 2) apothem

a point in the interior that is equidistant from all the vertices. Every regular polygon has a ______, center A segment drawn from the center that is perpendicular to a side of the regular polygon is called an ________. apothem In any regular polygon, all apothems are _________. congruent

Now, create a triangle by drawing segments from the center to each vertex on either side of the apothem. Now multiply this times the number of triangles that make up the regular polygon. The figure below shows a center and all vertices of a regular pentagon. The area of a triangle is calculated with the following formula: An apothem is drawn from the center, and is _____________ to a side. perpendicular There are 5 vertices and each is 72° from the other (360 ÷ 5 = ___.) 72 72° 72° 72° 72° s a 72° What measure does 5s represent? perimeter Rewrite the formula for the area of a pentagon using P for perimeter.

Theorem 10-5 Area of a Regular Polygon If a regular polygon has an area of A square units, an apothem of a units, and a perimeter of P units, then P

Find the area of the shaded region in the regular polygon. 5.5 ft 8 ft Area of polygon Area of triangle To find the area of the shaded region, subtract the area of the _______ from the area of the ________: triangle pentagon The area of the shaded region: 110 ft2 – 22 ft2 = 88 ft2

Find the area of the shaded region in the regular polygon. Area of polygon Area of triangle 6.9 m 8 m To find the area of the shaded region, subtract the area of the _______ from the area of the ________: triangle hexagon The area of the shaded region: 165.6 m2 – 55.2 m2 = 110.4 m2

End of Lesson

Circles § 11.1 Parts of a Circle § 11.2 Arcs and Central Angles
§ Arcs and Chords § Inscribed Polygons § Circumference of a Circle § Area of a Circle 491

What You'll Learn Vocabulary Parts of a Circle
You will learn to identify and use parts of circles. Vocabulary 1) circle 2) center 3) radius 4) chord 5) diameter 6) concentric 492

A circle is a special type of geometric figure.
Parts of a Circle A circle is a special type of geometric figure. All points on a circle are the same distance from a ___________. center point O B A The measure of OA and OB are the same; that is, OA  OB 493

Parts of a Circle Definition of a Circle A circle is the set of all points in a plane that are a given distance from a given _____ in the plane, called the ______ of the circle. point center P Note: a circle is named by its center. The circle above is named circle P. 494

There are three kinds of segments related to circles.
Parts of a Circle There are three kinds of segments related to circles. A ______ is a segment whose endpoints are the center of the circle and a point on the circle. radius A _____ is a segment whose endpoints are on the circle. chord A ________ is a chord that contains the center diameter 495

that passes through the center. chord
Parts of a Circle Segments of Circles radius chord diameter R K J K A K T G From the figures, you can not that the diameter is a special type of _____ that passes through the center. chord 496

Use Q to determine whether each statement is true or false.
Parts of a Circle Use Q to determine whether each statement is true or false. False; C A D B Q Segment AD does not go through the center Q. True; False; 497

All radii of a circle are _________. congruent
Parts of a Circle Theorem 11-1 11-2 All radii of a circle are _________. congruent P R S T G The measure of the diameter d of a circle is twice the measure of the radius r of the circle. 498

Parts of a Circle Find the value of x in Q. 3x – 5 = 2(17) 3x – 5 = 34
B Find the value of x in Q. 3x – 5 = 2(17) A 3x-5 17 3x – 5 = 34 Q 3x = 39 x = 13 C 499

P has a radius of 5 units, and T has a radius of 3 units.
Parts of a Circle P has a radius of 5 units, and T has a radius of 3 units. R P Q T B A If QR = 1, find RT RT = TQ – QR RT = 3 – 1 RT = 2 If QR = 1, find PQ If QR = 1, find AB PQ = PR – QR AB = 2(AP) + 2(BT) – 1 PQ = 5 – 1 AB = 2(5) + 2(3) – 1 PQ = 4 AB = – 1 AB = 15 500

Parts of a Circle P has a radius of 5 units, and T has a radius of 3 units. R P Q T B A If AR = 2x, find AP in terms of x. AR = 2(AP) 2x = 2(AP) x = AP If TB = 2x, find QB in terms of x. QB = 2(TB) QB = 2(2x) QB = 4x 501

Because all circles have the same shape, any two circles are similar.
Parts of a Circle Because all circles have the same shape, any two circles are similar. However, two circles are congruent if and only if (iff) their radii are _________. congruent Two circles are concentric if they meet the following three requirements: They lie in the same plane. S They have the same center. R T They have radii of different lengths. Circle R with radius RT and circle R with radius RS are concentric circles. 502

Arcs and Central Angles
What You'll Learn You will learn to identify major arcs, minor arcs, and semicircles and find the measures of arcs and central angles. Vocabulary 1) central angle 2) arcs 3) minor arc 4) major arc 5) semicircle 6) adjacent arc 503

A ____________ is formed when the two sides of an angle meet at the center of a circle. central angle Each side intersects a point on the circle, dividing it into ____ that are curved lines. arcs There are three types of arcs: T S A _________ is part of the circle in the interior of the central angle with measure less than 180°. minor arc central angle R A _________ is part of the circle in the exterior of the central angle. major arc Semicircles __________ are congruent arcs whose endpoints lie on the diameter of the circle. 504

Types of Arcs minor arc PG major arc PRG semicircle PRT K T P R G K P G K P G R Note that for circle K, two letters are used to name the minor arc, but three letters are used to name the major arc and semicircle. These letters for naming arcs help us trace the set of points in the arc. In this way, there is no confusion about which arc is being considered. 505

Depending on the central angle, each type of arc is measured in the following way. Definition of Arc Measure 1) The degree measure of a minor arc is the degree measure of its central angle. 2) The degree measure of a major arc is 360 minus the degree measure of its central angle. 3) The degree measure of a semicircle is 180. 506

In P, find the following measures: = APM P H A M T 46° 80° = 46° APT APT = 80° = 360° – (MPA + APT) = 360° – (46° + 80°) = 360° – (126°) = 234° 507

In P, AM and AT are examples of ________ arcs. adjacent P H A M T 46° 80° Adjacent arcs have exactly one point in common. For AM and AT, the common point is __. A Adjacent arcs can also be added. Postulate 11-1 Arc Addition The sum of the measures of two adjacent arcs is the measure of the arc formed by the adjacent arcs. C Q P R If Q is a point of PR, then mPQ + mQR = mPQR 508

In P, RT is a diameter. Find mQT. mQT + mQR = mTQR R P S Q T 75° 65° mQT ° = 180° mQT = 105° Find mSTQ. mSTQ + mQR + mRS = 360° mSTQ ° + 65° = 360° mSTQ ° = 360° mSTQ = 220° 509

Suppose there are two concentric circles with ASD forming two minor arcs, BC and AD. 60° A D B Are the two arcs congruent? S C Although BC and AD each measure 60°, they are not congruent. The arcs are in circles with different radii, so they have different lengths. However, in a circle, or in congruent circles, two arcs are congruent if they have the same measure. 510

Theorem 11-3 In a circle or in congruent circles, two minor arcs are congruent if and only if (iff) their corresponding central angles are congruent. Z Y X W Q WX  YZ iff 60° 60° mWQX = mYQZ 511

In M, WS and RT are diameters, mWMT = 125, mRK = 14. Find mRS. WMT  RMS Vertical angles are congruent T W S K M R WMT = RMS Definition of congruent angles mWT = mRS Theorem 11-3 125 = mRS Substitution Find mKS. Find mTS. KS + RK = RS TS + RS = 180 KS = 125 TS = 180 KS = 111 TS = 55 512

Twenty-two percent of all teens ages 12 through 17 work either full or part-time. Source: ICRs TeenEXCEL survey for Merrill Lynch Teens at Work The circle graph shows the number of hours they work per week. Find the measure of each central angle. 1 – 5: = 72 20% of 360 6 – 10: = 83 23% of 360 11 – 20: = 119 33% of 360 21 – 30: = 50 14% of 360 30 or More: = 36 10% of 360 513

What You'll Learn Vocabulary Arcs and Chords
You will learn to identify and use the relationships among arcs, chords, and diameters. Vocabulary Nothing New! 514

In circle P, each chord joins two points on a circle.
Arcs and Chords In circle P, each chord joins two points on a circle. Between the two points, an arc forms along the circle. By Theorem 11-3, AD and BC are congruent because their corresponding central angles are _____________, and therefore congruent. vertical angles P A C By the SAS Theorem, it could be shown that ΔAPD  ΔCPB. Therefore, AD and BC are _________. congruent B D The following theorem describes the relationship between two congruent minor arcs and their corresponding chords. S 515

In a circle or in congruent circles, two minor arcs are congruent
Arcs and Chords Theorem 11-4 In a circle or in congruent circles, two minor arcs are congruent if and only if (iff) their corresponding ______ are congruent. chords A D AD  BC iff B AD  BC C 516

The vertices of isosceles triangle ABC are located on R.
Arcs and Chords The vertices of isosceles triangle ABC are located on R. If BA  AC, identify all congruent arcs. A BA  AC R C B 517

Hands-On Arcs and Chords Step 1) Use a compass to draw circle on a
G H E F Step 1) Use a compass to draw circle on a piece of patty paper. Label the center P. Draw a chord that is not a diameter. Label it EF. P Step 2) Fold the paper through P so that E and F coincide. Label this fold as diameter GH. Q1: When the paper is folded, how do the lengths of EG and FG compare? EG  FG Q2: When the paper is folded, how do the lengths of EH and FH compare? EG  FG Q3: What is the relationship between diameter GH and chord EF? They appear to be perpendicular. 518

Arcs and Chords Theorem 11-5 In a circle, a diameter bisects a chord and its arc if and only if (iff) it is perpendicular to the chord. P R D C B A AR  BR and AD  BD iff CD AB Like an angle, an arc can be bisected. 519

Find the measure of AB in K.
Arcs and Chords Find the measure of AB in K. Theorem 11-5 Substitution B C A K D 7 520

Find the measure of KM in K if ML = 16.
Arcs and Chords Find the measure of KM in K if ML = 16. Pythagorean Theorem Given; Theorem 11-5 M K L N 6 521

What You'll Learn Vocabulary Inscribed Polygons
You will learn to inscribe regular polygons in circles and explore the relationship between the length of a chord and its distance from the center of the circle. Vocabulary 1) circumscribed 2) inscribed 522

Definition of Inscribed Polygon
Inscribed Polygons When the table’s top is open, its circular top is said to be ____________ about the square. circumscribed We also say that the square is ________ in the circle. inscribed Definition of Inscribed Polygon A polygon is inscribed in a circle if and only if every vertex of the polygon lies on the circle. 523

Some regular polygons can be
Inscribed Polygons Some regular polygons can be constructed by inscribing them in circles. B A Inscribe a regular hexagon, labeling the vertices, A, B, C, D, E, and F. C P F Construct a perpendicular segment from the center to each chord. From our study of “regular polygons,” we know that the chords AB, BC, CD, DE, and EF are _________ D E congruent From the same study, we also know that all of the perpendicular segments, called ________, are _________. apothems congruent The chords are congruent because the distances from the center of the circle are congruent. Make a conjecture about the relationship between the measure of the chords and the distance from the chords to the center. 524

In a circle or in congruent circles, two chords are congruent
Inscribed Polygons Theorem 11-6 In a circle or in congruent circles, two chords are congruent if and only if they are __________ from the center. equidistant B L M AD  BC iff P A LP  PM C D 525

In circle O, O is the midpoint of AB.
Inscribed Polygons O S T B A C R In circle O, O is the midpoint of AB. If CR = -3x and ST = 4x, find x definition of midpoint Theorem 11-6 substitution 526

Circumference of a Circle
What You'll Learn You will learn to solve problems involving circumferences of cirlces. Vocabulary 1) circumference 2) pi (π) 527

An in-line skate advertises “80-mm clear wheels.” The description “80-mm” refers to the diameter of the skates’ wheels. As the wheels of an in-line skate complete one revolution, the distance traveled is the same as the circumference of the wheel. Just as the perimeter of a polygon is the distance around the polygon, the circumference of a circle is the ______________________. distance around the circle 528

On a sheet of paper, create a table similar to the one below: Circumference Diameter Ratio: C/D Result Example Data 271 86 271 ÷ 86 Go to this website to Collect Data from various Circles. Go to this website to Analyze your Data. 529

In the previous activity, the ratio of the circumference C of a circle to its diameter d appears to be a fixed number slightly greater than 3, regardless of the size of the circle. The ratio of the circumference of a circle to its diameter is always fixed and equals an irrational number called __ or __. pi π Thus, ____ = __, π 530

Theorem 11-7 Circumference of a Circle If a circle has a circumference of C units and a radius of r units, then C = ____ or C = ___ d r C 531

What You'll Learn Vocabulary Area of a Circle
You will learn to solve problems involving areas and sectors of circles. Vocabulary 1) sector 532

The space enclosed inside a circle is its area.
Area of a Circle The space enclosed inside a circle is its area. By slicing a circle into equal pie-shaped pieces as shown below, you can rearrange the pieces into an approximate rectangle. Note that the length along the top and bottom of this rectangle equals the _____________ of the circle, ____. circumference So, each “length” of this approximate rectangle is half the circumference, or __ 533

Therefore, the area of this approximate rectangle is (π r)r or ___.
Area of a Circle The “width” of the approximate rectangle is the radius r of the circle. Recall that the area of a rectangle is the product of its length and width. Therefore, the area of this approximate rectangle is (π r)r or ___. 534

If a circle has an area of A square units and a radius of
Area of a Circle Theorem 11-8 Area of a Circle If a circle has an area of A square units and a radius of r units, then A = ___ r 535

You could calculate the area if you only knew the radius.
Area of a Circle Find the area of the circle whose circumference is meters. Round to the nearest hundredth. Use your knowledge of circumference. Solve for radius, r. You could calculate the area if you only knew the radius. Any ideas? 536

Area of a Sector of a Circle
Area of a Circle Theorem 11-9 Area of a Sector of a Circle If a sector of a circle has an area of A square units, a central angle measurement of N degrees, and a radius of r units, then 537

Surface Area and Volume
§ Solid Figures § Surface Areas of Prisms and Cylinders § Volumes of Prisms and Cylinders § Surface Areas of Pyramids and Cones § Volumes of Pyramids and Cones § Spheres § Similarity of Solid Figures

Right Triangles and Trigonometry
§ Simplifying Square Roots § ° - 45° - 90° Triangles § ° - 60° - 90° Triangles § Tangent Ratio § Sine and Cosine Ratios

Simplifying Radical Expressions
What You'll Learn You will learn to ______, _____, and _______ radical expressions. multiply divide simplify Vocabulary 1) perfect square 2) square root 3) radical sign 4) radical expression 5) radicand 6) simplest form

What You'll Learn Vocabulary You will learn to _________________
45° - 45° - 90° Triangles What You'll Learn You will learn to _________________ 45° – 45° – 90° triangles. use the properties of Vocabulary Nothing New!

Tangent Ratio What You'll Learn You will learn to use the __________ to solve problems. tangent ratio Vocabulary 1) trigonometry 2) trigonometric ratio 3) tangent 4) angle of elevation 5) angle of depression

Sine and Cosine Ratios What You'll Learn You will learn to use the _________________ to solve problems. sine and cosine ratios Vocabulary 1) sine 2) cosine 3) trigonometric identities

Circle Relationships § 14.1 Inscribed Angles
§ Tangents to a Circle § Secant Angles § Secant - Tangent Angles § Secant Measures § Equations of Circles

Simplifying Radical Expressions
What You'll Learn You will learn to ______, _____, and _______ radical expressions. multiply divide simplify Vocabulary 1) perfect square 2) square root 3) radical sign 4) radical expression 5) radicand 6) simplest form

Tangent Ratio What You'll Learn You will learn to use the __________ to solve problems. tangent ratio Vocabulary 1) trigonometry 2) trigonometric ratio 3) tangent 4) angle of elevation 5) angle of depression

Sine and Cosine Ratios What You'll Learn You will learn to use the _________________ to solve problems. sine and cosine ratios Vocabulary 1) sine 2) cosine 3) trigonometric identities

Reasoning in Geometry § 1.1 Patterns and Inductive Reasoning

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Reasoning in Geometry § 1.1 Patterns and Inductive Reasoning

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