Chapter 1 Tools of Geometry

1-1 Patterns and Inductive Reasoning 1-2 Points, Lines, and Planes 1-3 Segments , Rays, Parallel Lines, and Planes 1-4 Measuring Segments and Angles 1-5 Basic Constructions 1-6 The Coordinate Plane 1-7 Perimeter, Circumference, and Area

Definitions Acute angle Angle bisector Perpendicular bisector Collinear points Perpendicular lines Congruent angles Plane Congruent segments Point Conjecture Postulate Coordinate Ray Coplanar Right angle Counterexample Segment Inductive reasoning Skew lines Line Space Obtuse angle Straight angle Parallel lines Parallel planes

1-1 Patterns and Inductive Reasoning Goal : To use inductive reasoning to make conjectures Example 1 Use inductive reasoning to find a pattern for each sequence. Use the pattern to show the next two terms in the sequence. a. 3,6,12,24,… b. 6 12 24 Each circle has one more segment through the center of the circle. 2 x 24 = 48 and 2 x 48 = 96

A, E, I, ?, ? Inductive Reasoning Reasoning based on patterns you observe. A, E, I, ?, ?

Make a conjecture about the sum of the first 50 odd numbers. What are these numbers? Perfect squares!!! = 1 1 + 3 1 + 3 + 5 1 + 3 + 5 + 7 = 1 = 4 = 9 = 16 What can we conclude if we were to keep going? The sum of the first 30 odd numbers is 30 squared or 900

Conjecture A conclusion reached by inductive reasoning Example: Can you make a conjecture about the next Batman movie based on clues from The Dark Knight? What characters may be in it? Why? You are using both inductive reasoning and making a conjecture!

The speed with which a cricket chirps is affected by the temperature The speed with which a cricket chirps is affected by the temperature. If you hear 20 cricket chirps in 14 seconds, what is the temperature? Chirps per 14 Seconds 5 chirps 45°F 10 chirps 55°F 15 chirps 65°F 75°F Jeff works out regularly. When he first start exercising, he could do 10 push-ups. After the first month he could do 14. After the second month he could do 19, and after the third month he could do 25. Predict the number of push-ups Jeff will be able to do after the fifth month of working out. How confident are you of your prediction? Explain. Time and maximum number are two factors in making the prediction only an estimate. 40

Since neither is true this is a counterexample. Find one counter example to show that the conjecture is false. The difference of two integers is less than either integer. -6 – (-4) < -6 False -6 – (-4) < -4 False Since neither is true this is a counterexample.

Homework p. 6 – 7: 1, 12,17,21,22,25,28,57,58

1-2 Points, Lines, and Planes Goal- To understand basic terms and postulates of geometry. t B Point A Space Line

A Point A location Has no size Represented by a small dot and is named by a capital letter. A

Space The set of all points

Line A series of points that extends in two opposite directions with no end. A line is named by any two points on the line, such as (read “line AB”) or by a single lower case letter such as t ( read “line t”).

Identifying Collinear Points a. Are points E,F, and C collinear? If so, name the line they lie on. b. Are points E,F, and D collinear? If so, name the line they lie on. C F m E P D l a. Points E,F, and D are collinear. They lie on line m. b. Points E,F, and D are not collinear.

Collinear Points Points that lie on the same line. Collinear rays would be rays that lie on the same line.

In the figure below, name three points that are collinear and three points that are not collinear. Y Z V X W

P coplanar Plane P A B C Plane ABC

Plane A flat surface that has no thickness. A plane contains many lines and extends without end in the directions of all its lines. You name a plane by either a single capital letter (that is not a point on the plane) or by AT LEAST three of its noncollinear points.

Noncollinear Points that DO NOT lie on the same plane.

Coplanar Points and lines that are in the same plane.

Naming a Plane a. What do the dotted lines in a picture like this one mean? Each surface of the ice cube represents part of a plane. Name the plane represented by the front of the ice cube. List three different names for the plane represented by the top of the ice cube. H G E F D C A B a. The object is 3D, and the dotted lines represent the sides you cannot see. b. Plane AEF, Plane AEB, and Plane ABFE (these are only a few of the names). c. Plane HEF, Plane HEFG, and Plane FGH (these are only a few of the names).

Homework P 13-14

1-2 Points, Lines, and Planes Part 2 A postulate or axiom is an accepted statement or fact.

Postulate 1-1 Through any two points there is exactly one line. t B A Line t is the only line that passes through points A and B. What does that mean for any two points?

C B D A E H Any two points make exactly one line! F

Postulate 1-2 If two lines intersect, then they intersect in exactly one point. B A C and intersect at C. D E

Postulate 1-3 If two planes intersect, then they intersect in exactly one line. R W S Plane RST and plane STW intersect at T

What is the intersection of plane HGFE and plane BCGF? Name two planes that intersect at H G F F E a. D C b. Plane ABF and Plane CBF (can be named in other ways) A B

Postulate 1-4 Through any three noncollinear points there is exactly one plane.

Using Postulate 1-4 Shade the plane that contains A,B, and C. Shade the plane that contains E,H, and C. H H G G F F E E D D C C B B A A

Homework P 13 – 15: 2, 11, 31, 44, 45

1-3 Segments, Rays, Parallel Lines, and Planes Goal- To identify segments and rays. Goal- To recognize parallel lines. Endpoints Y Ray A B X Segment Q R S Opposite Rays

Segment The part of a line consisting of two endpoints and all the points between them. You name a segment by it’s endpoints with a line over it. Ex.

Ray The part of a line consisting of one endpoint and all the points of the line on one side of the endpoint. The endpoint must be the first letter!!! B This is ray A

Opposite Rays Two collinear rays with the same endpoint. Opposite rays ALWAYS form a line.

Naming Segments and Rays Name the segments and rays in the figure. Q P L

Parallel Lines Coplanar lines that do not intersect.

Skew Lines Noncoplanar, therefore, they ARE NOT parallel and DO NOT intersect.

E H G F A B C D and are Parallel or Skew Lines? and are Parallel or Skew Lines?

Parallel Planes Planes that do not intersect. In the classroom what planes are parallel? Can you think of other parallel planes?

Homework P 19 – 22 # 5, 16,17,40,41,50,54,55-60,75,81,83

1-4 Measuring Segments and Angles Goal: To find the lengths of segments To find the measures of angles

Ruler Postulate The distance between any two points is the absolute value of the difference of the two numbers. 2 4 6 8 -8 -6 -4 -2 AB = Length of = 6

Congruent Segments AB = CD We show two things are congruent with the following symbol: A B A B 2cm C D C D 2cm AB = CD

A B C D -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 AC = DB =

Segment Addition Postulate B C AB + BC = AC

A B C If AB = 5 and BC = 9, then AC = ??? b. If BC = 7 and AC = 20, then AB = ???

A B C 2x+3 4x-20 AC = 100

C is the midpoint of . Find AC, CB, and AB. 2x + 1 3x - 4

Midpoint A point that divides a segment into two congruent segments.

Angle Formed by 2 rays with the same endpoint. Rays are the sides of the angle. The endpoint is the vertex of the angle. Expressed by:

Naming an Angle Possible names for the angle: X Vertex Possible names for the angle: X, 1, or (with the vertex in the middle) AXB, and BXA 1 A B

One way to measure an angle is in degrees. To indicate the size or degree of an angle write a lowercase m in front of the angle symbol. Q m X = 78° Name angle X in two other ways: 78° P X

Classifying Angles x° x° Acute Angle Right Angle 0 < x < 90 Straight Angle X = 180 Obtuse Angle 90 < x < 180

Angle Addition Postulate B B A C O O C m AOC + m BOC = m AOC m AOC + m BOC = 180

Using the Angle Addition Postulate What is m TSW if m RST = 50 and m RSW = 125 W T R S

Marking Congruent Angles X A A B

Homework P 25-28 # 5,10,14,31-33,34,37,48,49

1-5 Basic Constructions A few things we need to know: Perpendicular Lines Perpendicular Bisector Angle Bisector

Perpendicular Lines Two lines that intersect to form right angles. The symbol means “ is perpendicular to”

Perpendicular Bisector A line, segment, or ray that is perpendicular to a segment at its midpoint.

Angle Bisector A ray that divides an angle into two angles.

1-6 The Coordinate Plane Goal: To find the distance between two points in the coordinate plane. Goal: To find the coordinate of the midpoint of a segment in the coordinate plane.

(x,y) describes the location of a point. J Quadrant II (-, +) Quadrant I (+, +) K L Quadrant III (-, -) Quadrant IV (+, -) M

The Distance Formula Given points and then: Memorize THIS!!!!!!!!

Finding Distance Find the distance between T(5,2) and R(-4,-1) to the nearest tenth.

Every morning Jordan takes the “Blue Line” subway from Sycamore Station to Byron Station. Sycamore Station is 1 mile West and 2 miles South of Emerald City. Byron Station is 2 miles East and 4 miles North of Emerald City. Find the distance Jordan travels between Sycamore Station and Byron Station? Byron Station Emerald City Sycamore Station

Midpoint Formula MEMORIZE THIS!!! The coordinates of the MIDPOINT M of a segment with endpoints and are: MEMORIZE THIS!!!

has endpoints Q(3,5) and S(7,-9) has endpoints Q(3,5) and S(7,-9). Find the coordinates of its midpoint M.

The midpoint of is M(3,4). One endpoint is A(-3,-2) The midpoint of is M(3,4). One endpoint is A(-3,-2). Find the coordinates of the other endpoint B.

Homework 1-5 P 33-36 # 8-10, 14, 15, 17, 35, 39, 40 1-6 P 40-42 # 2, 5, 9, 12, 16, 19, 31, 32,46

1-7 Perimeter, Circumference, and Area Goal: To find the perimeters of rectangles and squares, and circumferences of circles. To find areas of rectangles, squares, and circles.

Perimeter The sum of the lengths of all the sides of a polygon. + + +

Perimeter P = 2b + 2h C = Circumference d = diameter r = radius C = d SQUARE RECTANGLE h h b b P = 2b + 2h C = Circumference d = diameter r = radius C = d C = 2r r CIRCLE d O

Your pool is 15 ft wide and 20 ft long with a 3 ft wide deck surrounding it. You want to build a fence around the deck. How much fencing will you need? 3 ft 20 ft 15 ft

Finding Circumference Find the circumference of circle A in terms of . Then find the circumference to the nearest tenth. 12 in A

Find the Perimeter of ABC. Y X A C B

You are designing a banner for homecoming You are designing a banner for homecoming. The size of the banner will be 6 ft wide and 8 ft high. How much material do you need?

The diameter of a circle is 10 in. Find the area in terms of .

Homework P 43 Project ( steps 1-5), P47 # 3,14

Postulate 1-9 Postulate 1-10 If two figures are , then their areas are equal. Postulate 1-10 The area of a region is the sum of the areas of its non-overlapping parts.

Find the Perimeter and Area of the figure.

Homework P 55 16-19, 41(a &b), 42

Homework P 65 -68 # 7, 17, 29, 51, 58, 66, 71, 72

END OF CHAPTER!!!!! Make sure to Review P 71-73 mult of 3 Complete the given study guide as well