Geometry 1 Unit 1: Basics of Geometry

1.1 Patterns and Inductive Reasoning Geometry 1 Unit 1 1.1 Patterns and Inductive Reasoning

EXAMPLE 1 Describe a visual pattern Describe how to sketch the fourth figure in the pattern. Then sketch the fourth figure. SOLUTION Each circle is divided into twice as many equal regions as the figure number. Sketch the fourth figure by dividing a circle into eighths. Shade the section just above the horizontal segment at the left.

GUIDED PRACTICE for Examples 1 and 2 Sketch the fifth figure in the pattern in example 1. ANSWER

EXAMPLE 2 Describe a number pattern Describe the pattern in the numbers –7, –21, –63, –189,… and write the next three numbers in the pattern. Notice that each number in the pattern is three times the previous number. Continue the pattern. The next three numbers are –567, –1701, and –5103. ANSWER

GUIDED PRACTICE for Examples 1 and 2 Describe the pattern in the numbers 5.01, 5.03, 5.05, 5.07,… Write the next three numbers in the pattern. 2. 5.13 Notice that each number in the pattern is increasing by 0.02. 5.11 +0.02 5.09 5.07 5.05 5.03 5.01 Continue the pattern. The next three numbers are 5.09, 5.11 and 5.13 ANSWER

Patterns and Inductive Reasoning Conjecture An unproven statement that is based on observations. Inductive Reasoning The process of looking for patterns and making conjectures.

EXAMPLE 3 Make a conjecture Given five collinear points, make a conjecture about the number of ways to connect different pairs of the points. SOLUTION Make a table and look for a pattern. Notice the pattern in how the number of connections increases. You can use the pattern to make a conjecture.

EXAMPLE 3 Make a conjecture Conjecture: You can connect five collinear points 6 + 4, or 10 different ways. ANSWER

EXAMPLE 4 Make and test a conjecture Numbers such as 3, 4, and 5 are called consecutive integers. Make and test a conjecture about the sum of any three consecutive integers. SOLUTION STEP 1 Find a pattern using a few groups of small numbers. 3 + 4 + 5 = 12 = 4 3 7 + 8 + 9 = 12 = 8 3 10 + 11 + 12 = 33 = 11 3 16 + 17 + 18 = 51 = 17 3 Conjecture: The sum of any three consecutive integers is three times the second number. ANSWER

EXAMPLE 4 Make and test a conjecture STEP 1 Test your conjecture using other numbers. For example, test that it works with the groups –1, 0, 1 and 100, 101, 102. –1 + 0 + 1 = 0 = 0 3 100 + 101 + 102 = 303 = 101 3

GUIDED PRACTICE for Examples 3 and 4 3. Make and test a conjecture about the sign of the product of any three negative integers. Test: Test conjecture using the negative integers –2, –5 and –4 –2 –5 –4 = –40 Conjecture: The result of the product of three negative numbers is a negative number. ANSWER

Patterns and Inductive Reasoning Counterexample An example that shows a conjecture is false.

EXAMPLE 5 Find a counterexample A student makes the following conjecture about the sum of two numbers. Find a counterexample to disprove the student’s conjecture. Conjecture: The sum of two numbers is always greater than the larger number. SOLUTION To find a counterexample, you need to find a sum that is less than the larger number.

EXAMPLE 5 Find a counterexample –2 + –3 = –5 –5 > –2 Because a counterexample exists, the conjecture is false. ANSWER

( )2 GUIDED PRACTICE for Examples 5 and 6 5. Find a counterexample to show that the following conjecture is false. Conjecture: The value of x2 is always greater than the value of x. 12 ( )2 = 14 14 > 12 Because a counterexample exist, the conjecture is false ANSWER

Unit 1-Basics of Geometry 1.2: Points, Lines and Planes

Points, Lines, and Planes Definition Uses known words to describe a new word. Undefined terms Words that lack a formal definition. In Geometry it is important to have a general agreement about these words. The building blocks of Geometry are undefined terms.

Points, Lines, and Planes The 3 Building Blocks of Geometry: Point Line Plane These are called the “building blocks of geometry” because these terms lay the foundation for Geometry.   These are “undefined” terms. They are not formally defined. We agree on a common description for each of these. A definition is a statement that clarifies the meaning of a word or phrase. It is impossible to define point, line and plane without using words that must also be defined, so they remain undefined.

Points, Lines, and Planes The most basic building block of Geometry Has no size A location in space Represented with a dot Named with a Capital Letter  

Points, Lines, and Planes Example: point P   P

Points, Lines, and Planes Set of infinitely many points One dimensional, has no thickness Goes on forever in both directions Named using any two points on the line with the line symbol over them, or a lowercase script letter

Points, Lines, and Planes Example: line AB, AB, BA or l B A   **2 points determine a line l

Points, Lines, and Planes Has length and width, but no thickness A flat surface that extends infinitely in 2-dimensions (length and width) Represented with a four-sided figure like a tilted piece of paper, drawn in perspective Named with a script capital letter or 3 points in the plane

Points, Lines, and Planes Example: Plane P or plane ABC A C B P **3 noncollinear points determine a plane

Points, Lines, and Planes Collinear Points that lie on the same line Points A, B, and C are Collinear A B C

Points, Lines, and Planes Coplanar Points that lie on the same plane Points D, E, and F are Coplanar D E F

Points, Lines, and Planes Line Segment Two points (called the endpoints) and all the points between them that are collinear with those two points Named line segment AB, AB, or BA line AB segment AB  A B A B Always name a segment using the endpoints.

Points, Lines, and Planes Ray Part of a line that starts at a point and extends infinitely in one direction. Initial Point Starting point for a ray. Ray CD, or CD, is part of CD that contains point C and all points on line CD that are on the same side as of C as D “It begins at C and goes through D and on forever”

Segments and Their Measures Between When three points are collinear, you can say that one point is between the other two. A B C D E F Point B is between A and C Point E is NOT between D and F

Points, Lines, and Planes Opposite Rays If C is between A and B, then CA and CB are opposite rays. Together they make a line. Use yarn and have students go to front and show this. A B C

Points, Lines, and Planes C Y D C Y D C Y D   Line CD Ray DC Ray CD CD and CY represent the same ray.  Notice CD is not the same as DC. ray CD is not opposite to ray DC

Points, Lines, and Planes The intersection of two lines is a point. The intersection of two planes is a line.

Unit 1-Basics of Geometry 1.3: Segments and Their Measures

Segments and Their Measures Postulates Rules that are accepted without proof. Also called axioms

Segments and Their Measures Ruler Postulate The points on a line can be matched one to one with the real numbers. The real number that corresponds to a point is called the coordinate of the point. The distance between points A and B, written as AB, is the absolute value of the difference between the coordinates of A and B. AB is also called the length of AB.

Segments and Their Measures Segment length can be given in several different ways. The following all mean the same thing. A to B equals 2 inches AB = 2 in. mAB = 2 inches If no measurement units are used then it is understood that the choice of units is not important. (If a grid is given, it is based on the smallest square in the grid.)

Segments and Their Measures Example 1 Measure the length of the segment to the nearest millimeter. D E

Segments and Their Measures Between When three points are collinear, you can say that one point is between the other two. A B C D E F Point B is between A and C Point E is NOT between D and F

Segments and Their Measures Segment Addition Postulate If B is between A and C, then AB + BC = AC. If AB + BC = AC, then B is between A and C. AC AB BC A B C Use yarn and have students go to front and show this.

Segments and Their Measures Example 2 Two friends leave their homes and walk in a straight line toward the others home. When they meet, one has walked 425 yards and the other has walked 267 yards. How far apart are their homes?

Segments and Their Measures The Distance Formula A formula for computing the distance between two points in a coordinate plane. If A(x1,y1) and B(x2,y2) are points in a coordinate plane, then the distance between A and B is

Segments and Their Measures Example 3 Find the lengths of the segments. Tell whether any of the segments have the same length.

Segments and Their Measures Congruent Two segments are congruent if and only if they have the same measure. The symbol for congruence is . We use = between equal numbers and  between congruent figures.

Segments and Their Measures Markings on figures are used to show congruence. Use identical markings for each pair of congruent parts.   A 2.5 B AB = DC = 2.5 AB  DC D 2.5 C AD  BC

Segments and Their Measures Distance Formula and Pythagorean Theorem (AB)2 = (x2 – x1)2 + (y2 – y1)2 c2 = a2 + b2 |x2 – x1| |y2 – y1| A(x1, y1) B(x2, y2) C(x2, y1) b c a

Segments and Their Measures Example 4 On the map, the city blocks are 410 feet apart east-west and 370 feet apart north south. Find the walking distance between C and D. What would the distance be if a diagonal street existed between the two points?

Unit 1-Basics of Geometry 1.4: Angles and Their Measures

Angles and Their Measures Formed by two rays that share a common endpoint. Sides The rays that make the angle. Vertex The initial point of the rays.

Angles and Their Measures When naming an angle, the vertex must be the middle letter. angle CAT, angle TAC, CAT or TAC C A T

Angles and Their Measures If a vertex has only one angle then you can name it with that letter alone. TAC could also be called A. C A T

Angles and Their Measures Example 1 Name all the angles in the following drawing B C A D 1

Angles and Their Measures Protractor Geometry tool used to measure angles. Angles are measured in Degrees. Things to know A full circle is 360 degrees, or 360º. A line is 180º.

Angles and Their Measures Measure of an Angle The smallest rotation between the two sides of the angle. Congruent angles Angles that have the same measure. Use identical markings to show angles are congruent.

Angles and Their Measures Angle measure notation Use an m before the angle symbol to show the measure: mA = 34º or measure of A = 34º

Angles and Their Measures Protractor Postulate Consider a point A not on OB. The rays of the form OA can be matched one to one with the real numbers from 0 to 180. The measure of an angle is equal to the number on the protractor which one side of the angle passes through when the other side goes through the number zero on the same scale. A O B

Angles and Their Measures Step 1: Place the center mark of the protractor on the vertex. Step 2: Line up the 0-mark with one side of the angle. Step 3: Read the measure on the protractor scale. **Be sure you are reading the scale with the 0-mark you are using.

Angles and Their Measures Interior A point is in the interior if it is between points that lie on each side of the angle. Exterior A point is in the exterior of an angle if it is not on the angle or in its interior. E D exterior interior

Angles and Their Measures Angle Addition Postulate If P is in the interior of RST, then mRSP + mPST = mRST R m RST m RSP S P m PST T

Angles and Their Measures Example 2 The backyard of a house is illuminated by a light fixture that has two bulbs. Each bulb illuminates an angle of 120°. If the angle illuminated only by the right bulb is 35°, what is the angle illuminated by both bulbs? 120-350= 85 Left only Right only Both bulbs

Angles and Their Measures Acute Angle An angle whose measure is greater than 0° and less than 90º.

Angles and Their Measures Right Angle An angle whose measure is 90º

Angles and Their Measures Obtuse Angle An angle whose measure is greater than 90º and less than 180º.

Angles and Their Measures Straight Angle An angle whose measure is 180°. A

Angles and Their Measures Example 3 Plot the following points. A(-3, -1), B(-1, 1), C(2, 4), D(2, 1), and E(2, -2) Measure and classify the following angles as acute, right, obtuse or straight. a. DBE b. EBC c. ABC d. ABD

Angles and Their Measures

Angles and Their Measures Adjacent Angles Angles that share a common vertex and side, but have no common interior points. C A B D

Angles and Their Measures Example 4 Use a protractor to draw two adjacent angles LMN and NMO so that LMN is acute and LMO is straight.

Unit 1-Basics of Geometry 1.5: Segment and Angle Bisectors

Segment and Angle Bisectors Midpoint The point on the segment that is the same distance from both endpoints. This point bisects the segment. Bisect To cut in half (two equal pieces).

Segment and Angle Bisectors M is the midpoint of LN L M N LM  MN

Segment and Angle Bisectors Segment bisector A segment, ray, line, or plane that intersects a segment at its midpoint.

Segment and Angle Bisectors Compass Geometric tool that is used to construct circles and arcs. Straightedge Ruler without marks. Construction Geometric drawing that uses a compass and straightedge.

Segment and Angle Bisectors Construct a Segment Bisector and Midpoint Use the following steps to construct a bisector of AB and find the midpoint M of AB. Place the compass point at A. Use a compass setting greater than half of AB. Draw an arc. Keep the same compass setting. Place the compass point at B. Draw an arc. It should intersect the other arc in two places. Use a straightedge to draw a segment through the points of intersection. This segment bisects AB at M, the midpoint of AB.

Segment and Angle Bisectors Midpoint Formula Given two points (x1, y1) and (x2, y2) the coordinates of the midpoint are: x1 + x2 , y1 + y2 2 2 Average the x values and average the y values

Segment and Angle Bisectors Example 1 Find the coordinates of the midpoint of the segment with endpoints at (12, -8) and (-3, 15).

Segment and Angle Bisectors Example 2 Find the coordinates of the midpoint of the segment with endpoints at (5, 8) and (7, -2).

Segment and Angle Bisectors Example 3 One endpoint is (17,-3) and the midpoint is (8,2). Find the coordinates of the other endpoint.

Segment and Angle Bisectors Example 4 One endpoint is (-5,8) and the midpoint is (6,3). Find the coordinates of the other endpoint.

Segment and Angle Bisectors A ray that divides an angle into two adjacent angles that are congruent. D C A B mACD = mBCD

Segment and Angle Bisectors Construct an Angle Bisector Place the compass point at C. Draw an arc that intersects both sides of the angle. Label the intersections A and B. Place the compass point at A. Draw another arc. Then place the compass point at B. Using the same compass setting, draw a third arc to intersect the second one. Label the intersection D. Use a straightedge to draw a ray from C through D. This is the angle bisector. Students can check to see that it is the angle bisector by measuring each angle or folding along the bisector.

Segment and Angle Bisectors Example 5 JK bisects HJL. Given that mHJL = 42°, what are the measures of HJK and KJL?

Segment and Angle Bisectors Example 6 A cellular phone tower bisects the angle formed by the two wires that support it. Find the measure of the angle formed by the two wires. wire 47° Cellular phone tower

Segment and Angle Bisectors Example 7 MO bisects LMN. The measures of the two congruent angles are (3x – 20)° and (x + 10) °. Solve for x.

Unit 1-Basics of Geometry 1.6 Angle Pair Relationships

Angle Pair Relationships Vertical Angles Angles whose sides form opposite rays. 1 and 3 are vertical angles. 2 and 4 are vertical angles. 1 4 2 3

Angle Pair Relationships Linear Pair of Angles Angles that share a common vertex and a common side. Their non-common sides form a line. 5 and 6 are a linear pair of angles. 5 6

Angle Pair Relationships Example 1 Are 1 and 2 a linear pair? Are 4 and 5 a linear pair? Are 5 and 3 vertical angles? Are 1 and 3 vertical angles? 1 2 3 4 5

Angle Pair Relationships Example 2

Angle Pair Relationships Example 3 Solve for x and y. Then find the angle measures. (4x + 15)° (5x + 30)° (3y – 15)° (3y + 15)° L M O N P

Angle Pair Relationships Complementary Angles Two angles that have a sum of 90º Each angle is a complement of the other.

Angle Pair Relationships Supplementary Angles Two angles that have a sum of 180º Each angle is a supplement of the other.

Angle Pair Relationships Example 4 State whether the two angles are complementary, supplementary or neither. The angles formed by the hands of a clock at 11:00 and 1:00.

Angle Pair Relationships Example 5 Given that G is a supplement of H and mG is 82°, find mH. Given that U is a complement of V, and mU is 73°, find mV.

Angle Pair Relationships Example 6 T and S are supplementary. The measure of T is half the measure of S. Find mS.

Angle Pair Relationships Example 7 D and E are complements and D and F are supplements. If mE is four times mD, find the measure of each of the three angles.

Unit 1-Basics of Geometry 1.7: Introduction to Perimeter, Circumference, and Area

Introduction to Perimeter, Circumference, and Area Square Side length s P = 4s A = s2 s

Introduction to Perimeter, Circumference, and Area Rectangle Length l and width w P = 2l + 2w A = lw l w

Introduction to Perimeter, Circumference, and Area Triangle Side lengths a, b, and c, Base b, and height h P = a + b + c A = ½bh a c h b

Introduction to Perimeter, Circumference, and Area Circle Radius r C = 2π r A = π r2 Pi (π) is the ratio of the circle’s circumference to its diameter. π ≈ 3.14 r

Introduction to Perimeter, Circumference, and Area Example 1 Find the perimeter and area of a rectangle of length 4.5m and width 0.5m.

Introduction to Perimeter, Circumference, and Area Example 2 A road sign consists of a pole with a circular sign on top. The top of the circle is 10 feet high and the bottom of the circle is 8 feet high. Find the diameter, radius, circumference and area of the circle. Use π ≈ 3.14.

Introduction to Perimeter, Circumference, and Area Example 3 Find the area and perimeter of the triangle defined by H(-2, 2), J(3, -1), and K(-2, -4).

Introduction to Perimeter, Circumference, and Area Example 4 A maintenance worker needs to fertilize a 9-hole golf course. The entire course covers a rectangular area that is approximately 1800 feet by 2700 feet. Each bag of fertilizer covers 20,000 square feet. How many bags will the worker need?

Introduction to Perimeter, Circumference, and Area Example 5 You are designing a mat for a picture. The picture is 8 inches wide and 10 inches tall. The mat is to be 2 inches wide. What is the area of the mat?

Introduction to Perimeter, Circumference, and Area Example 6 You are making a triangular window. The height of the window is 18 inches and the area should be 297 square inches. What should the base of the window be?