Chapter 1: Tools of Geometry

Name: Chapter 1: Tools of Geometry
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Description: Chapter 1: Tools of Geometry

Chapter 1: Tools of Geometry
What geometric terms are you familiar with?

Patterns and Inductive Reasoning
Lesson 1-1 Patterns and Inductive Reasoning Check Skills You’ll Need (For help, go the Skills Handbook, page 753.) Here is a list of the counting numbers: 1, 2, 3, 4, 5, . . . Some are even and some are odd. 1. Make a list of the positive even numbers. 2. Make a list of the positive odd numbers. 3. Copy and extend this list to show the first 10 perfect squares. 12 = 1, 22 = 4, 32 = 9, 42 = 16, . . . 4. Which do you think describes the square of any odd number? It is odd. It is even.

Lesson 1-1 Patterns and Inductive Reasoning To use inductive reasoning to make conjectures. Objective

Lesson 1-1 Patterns and Inductive Reasoning Key Concepts Inductive reasoning is A conjecture is A counterexample to a conjecture is

Lesson 1-1 Patterns and Inductive Reasoning Example 1 Find a pattern for the sequence. Use the pattern to show the next two terms in the sequence. 384, 192, 96, 48, …

Lesson 1-1 Patterns and Inductive Reasoning Example 2 Make a conjecture about the sum of the cubes of the first 25 counting numbers. Find the first few sums. Notice that each sum is a perfect square and that the perfect squares form a pattern. 13 = 1 = 12 = 12 = 9 = 32 = (1 + 2)2 = 36 = 62 = ( )2 = 100 = 102 = ( )2 = 225 = 152 = ( )2

Lesson 1-1 Patterns and Inductive Reasoning Example 3 Find a counterexample for each conjecture. a. A number is always greater than its reciprocal. b. If a number is divisible by 5, then it is divisible by 10.

Lesson 1-1 Patterns and Inductive Reasoning Example 4 The price of overnight shipping was $8.00 in 2000, $9.50 in 2001, and $11.00 in Make a conjecture about the price in 2003.

Lesson 1-1 Patterns and Inductive Reasoning Lesson Quiz Find a pattern for each sequence. Use the pattern to show the next two terms or figures. 1. 3, –6, 18, –72, 360 2.

Lesson 1-1 Patterns and Inductive Reasoning Lesson Quiz (continued) Use the table and inductive reasoning. 3. Find the sum of the first 10 counting numbers. 4. Find the sum of the first 1000 counting numbers. Show that the conjecture is false by finding one counterexample. 5. The sum of two prime numbers is an even number.

Lesson 1-1 Patterns and Inductive Reasoning Homework Pages 6-8 19-28, even

Drawing, Nets, and Other Models
Lesson 1-2 Drawing, Nets, and Other Models Check Skills You’ll Need (For help, go to Lesson 1-1.) Draw the next figure in each sequence. 1. 2.

Lesson 1-2 Drawing, Nets, and Other Models To make isometric and orthographic drawings. Objective 1 To draw nets for three-dimensional figures. Objective 2 13

Lesson 1-2 Drawing, Nets, and Other Models Key Concepts An isometric drawing of a three-dimensional object An orthographic drawing is A foundation drawing shows A net is 14

Lesson 1-2 Drawing, Nets, and Other Models Example 1 Make an isometric drawing of the cube structure below.

Lesson 1-2 Drawing, Nets, and Other Models Example 2 Make an orthographic drawing of the isometric drawing below.

Lesson 1-2 Drawing, Nets, and Other Models Example 3 Create a foundation drawing for the isometric drawing below.

Lesson 1-2 Drawing, Nets, and Other Models Is the pattern a net for a cube? If so, name two letters that will be on opposite faces. Example 4

Lesson 1-2 Drawing, Nets, and Other Models Draw a net for the figure with a square base and four isosceles triangle faces. Label the net with its dimensions. Example 5

Lesson 1-2 Drawing, Nets, and Other Models Lesson Quiz Use the figure at the right for Exercises 1–2. 1. Make an isometric drawing of the cube structure. 2. Make an orthographic drawing.

Lesson 1-2 Drawing, Nets, and Other Models Lesson Quiz (continued) 3. Is the pattern a net for a cube? If so, name two letters that will be on opposite faces. 4. Draw a net for the figure.

Lesson 1-2 Drawing, Nets, and Other Models Homework Pages 13-14 1-16, 18-20, 23-26

Lesson 1-3 Points, Lines, and Planes
Check Skills You’ll Need (For help, go to the Skills Handbook, page 760.) 1. y = x y = 2x – 4 3. y = 2x y = –x y = 4x – y = –x + 15 4. Copy the diagram of the four points A, B, C, and D. Draw as many different lines as you can to connect pairs of points. Solve each system of equations.

To understand the basic terms of Geometry Objective 1 To understand the basic postulates of Geometry Objective 2

Key Concepts Three basic undefined terms: point line plane Point A point is a location or position. A point has no size. It is represented by a small dot and is named by a capital letter. A geometric figure is a set of points. Space is defined as the set of all points.

Key Concepts Line A line is You can name a line by any two points on the line or by a single lowercase script letter Points that lie on the same line are t SG or GS or line t

Key Concepts Plane A plane is You can name a plane by a single capital letter. Planes can also be named by at least three of its noncollinear points. Points and lines in the same plane are P Plane P Plane ABC

Example 1 In the figure below, name three points that are collinear and three points that are not collinear.

Example 2 Name the plane shown in two different ways.

C D A B F G H Example 3 a) Name two different planes that contain points C and G. b) Name all the planes that contain point E.

Key Concepts A postulate or axiom is Postulate: Through any two points there is exactly one line. t Line t is the only line that passes through points A and B

Key Concepts Postulate: If two lines intersect, then they intersect in exactly one point. C Lines AE and BD intersect at C

Key Concepts Postulate: If two planes intersect, then they intersect in exactly one line. N M Plane M and plane N intersect in RS

Key Concepts Postulate: Through any three noncollinear points there is exactly one plane.

Example 4 Use the diagram below. What is the intersection of plane HGC and plane AED?

Lesson Quiz Use the diagram at right. 1. Name three collinear points. 2. Name two different planes that contain points C and G. 3. Name the intersection of plane AED and plane HEG. 4. How many planes contain the points A, F, and H? 5. Show that this conjecture is false by finding one counterexample: Two planes always intersect in exactly one line.

Homework Pages 19 – 21; 1 – 24, 30 – 52 even, 55 – 60

Lesson 1-4 Segments, Rays, Parallel Lines, and Planes
Check Skills You’ll Need (For help, go to Lesson 1-3.) 4. the bottom 5. the top 6. the front 7. the back 8. the left side 9. the right side Judging by appearances, will the lines intersect? Name the plane represented by each surface of the box.

To identify segments and rays. Objective 1 To recognize parallel lines. Objective 2

Key Concepts A line segment is A B Segment AB

Key Concepts A ray is Y Ray YX

Key Concepts Opposite rays are RQ and RS are opposite rays.

Example 1 Name the segments and rays in the figure.

Key Concepts Parallel lines are Skew lines

Example 2 Use the figure below. Name all segments that are parallel to AE. Name all segments that are skew to AE.

Example 3 Identify a pair of parallel planes in your classroom.

Lesson Quiz Use the figure below for Exercises 1-3. 1. Name the segments that form the triangle. 2. Name the rays that have point T as their endpoint. 3. Explain how you can tell that no lines in the figure are parallel or skew.

Lesson Quiz (continued) Use the figure below for Exercises 4 and 5. 4. Name a pair of parallel planes. 5. Name a line that is skew to XW.

Homework Pages 25-27 1-35, 39, 41-45 Quiz 1-1 through 1-4 Friday, Sept. 9

Lesson 1-5 Measuring Segments
Check Skills You’ll Need (For help, go to the Skills Handbook, pages 757 and 758.) Simplify each absolute value expression. 1. |–6| 2. |3.5| 3. |7 – 10| 4. |–4 – 2| 5. |–2 – (–4)| 6. |–3 + 12| 7. x + 2x – 6 = 6 8. 3x + 9 + 5x = 81 9. w – 2 = –4 + 7w Solve each equation.

Finding segment lengths. Objective

Key Concepts Ruler Postulate: The points of a line can be put into one-to-one correspondence with the real numbers so that the distance between any two points is the absolute value of the difference of the corresponding numbers. the length of AB A B a b AB = coordinate of A coordinate of B

Key Concepts Congruent () segments are A B C D 2 cm AB = CD

Example 1 Find which two of the segments XY, ZY, and ZW are congruent.

Key Concepts Segment Addition Postulate: If three points A, B, and C, are collinear and B is between A and C, then AB + BC = AC. A C B ●

Example 2 If AB = 25, find the value of x. Then find AN and NB.

Key Concepts A midpoint of a segment is A C B ●

Example 3 M is the midpoint of RT. Find RM, MT, and RT.

Lesson Quiz Use the figure below for Exercises 1–3. 1. If XT = 12 and XZ = 21, then TZ = 2. If XZ = 3x, XT = x + 3, and TZ = 13, find XZ. 3. Suppose that T is the midpoint of XZ. If XT = 2x + 11 and XZ = 5x + 8, find the value of x.

Homework Pages 33 – 34 1 – 12, 14 – 22 even, 30 – 36 even

Lesson 1-6 Measuring Angles
Check Skills You’ll Need (For help, go to the Skills Handbook, pages 758.) Solve each equation. a = 130 2. m – 110 = 20 3. 85 – n = 40 4. x + 45 = 180 5. z – 20 = 90 – y = 135

Finding angle measures. Objective 1 Identifying angle pairs. Objective 2

Key Concepts An angle () is formed by The rays are the of the angle. The endpoint is the of the angle. 1 B 1, B, ABC, CBA

Example 1 Name the angle below in four ways.

Example 2 B 1 2 a) Name 1 in two other ways. b) Name DBC in two other ways.

Key Concepts You can classify angles according to their measures. acute angle right angle 0 < x < 90 x = 90 obtuse angle straight angle 90 < x < x = 180 x° x° x° x°

Key Concepts Angle Addition Postulate: If point B is in the interior of AOC, then mAOB + mBOC = mAOC. O O If AOC is a straight angle, then mAOB + mBOC = 180.

Example 3 Suppose that m1 = 42 and mABC = 88. Find m2.

Key Concepts Angles with the same measure are congruent angles. If m1 = m2, then 1 2. Angles can be marked alike to show they are congruent. A B C D E F

Key Concepts Vertical angles Adjacent angles Complementary angles Supplementary angles

Example 4 Name all pairs of angles in the diagram that are: a. vertical b. supplementary

Example 4 (continued) c. complementary

Example 5 Use the diagram below. Which of the following can you conclude: 3 is a right angle, 1 and 5 are adjacent, 3 @ 5?

Lesson Quiz 1. Name 2 two different ways. 2. Measure and classify 1, 2, and BAC.

Lesson Quiz (continued) Use the figure below for Exercises 3–4. 3. Name a pair of supplementary angles. 4. Can you conclude that there are vertical angles in the diagram? Explain.

Homework Page 40 2 – 34 even, 42 – 47 Quiz 1-5 & 1-6 Friday 9/16

Lesson 1-7 Basic Constructions
Check Skills You’ll Need (For help, go to Lessons 1-5 and 1-6.) 1. CD 2. GH 3. AB 4. line m 5. acute ABC 6. XY || ST 7. DE = 20. Point C is the midpoint of DE. Find CE. 8. Use a protractor to draw a 60° angle. 9. Use a protractor to draw a 120° angle. In Exercises 1-6, sketch each figure.

Use a compass and straight edge to construct congruent segments and rays. Objective 1 Use a compass and straight edge to bisect segments and angles. Objective 2 78

Example 1 Construct TW congruent to KM.

Example 2 Construct Y so that Y  G.

Key Concepts Perpendicular lines () are A perpendicular bisector of a segment is 81

Example 3 Construct the perpendicular bisector of AB.

Example 4 WR bisects AWB. mAWR = x and mBWR = 4x – 48. Find mAWB.

Example 5 Construct MX, the bisector of M.

Lesson Quiz Use the figure at right. NQ bisects DNB. 1. Construct AC so that AC  NB. 2. Construct the perpendicular bisector of AC. 3. Construct RST so that RST  QNB. 4. Construct the bisector of RST. 5. Find x. 6. Find mDNB.

Homework Pages 47 – 49 9 – 12 , 21, 25, 36

Lesson 1-8 The Coordinate Plane
Check Skills You’ll Need (For help, go to the Skills Handbook, pages 753 and 754.) Find the square root of each number. Round to the nearest tenth if necessary. 4. (m – n)2 5. (n – m)2 6. m2 + n2 7. (a – b) Evaluate each expression for m = –3 and n = 7. Evaluate each expression for a = 6 and b = –8. a + b 2 a2 + b2

Finding distance on the coordinate plane. Objective 1 Finding the midpoint of a segment. Objective 2

Key Concepts The coordinate plane

Key Concepts The Distance Formula: The distance d between two points A(x1, y1) and B(x2, y2) is

Example 1 Find the distance between R(–2, –6) and S(6, –2) to the nearest tenth.

Example 2 How far is the subway ride from Oak to Symphony? Round to the nearest tenth.

Key Concepts The Midpoint Formula: The coordinates of the midpoint M of AB with endpoints A(x1, y1) and B(x2, y2) are the following: The coordinates of the midpoint of a segment is (average of x-coordinates, average of y-coordinates)

Example 3 AB has endpoints (8, 9) and (–6, –3). Find the coordinates of its midpoint M.

Example 4 The midpoint of DG is M(–1, 5). One endpoint is D(1, 4). Find the coordinates of the other endpoint G.

Lesson Quiz A has coordinates (3, 8). B has coordinates (0, –4). C has coordinates (–5, –6). 1. Find the distance between A and B to the nearest tenth. 2. Find BC to the nearest tenth. 3. Find the midpoint M of AC to the nearest tenth. 4. B is the midpoint of AD. Find the coordinates of endpoint D. 5. An airplane flies from Stanton to Mercury in a straight flight path. Mercury is 300 miles east and 400 miles south of Stanton. How many miles is the flight? 6. Toni rides 2 miles north, then 5 miles west, and then 14 miles south. At the end of her ride, how far is Toni from her starting point, measured in a straight line?

Homework Page 56 2 – 42 even Chapter 1 Test Friday, 9/23

Lesson 1-9 Perimeter, Circumference, and Area
Check Skills You’ll Need (For help, go to the Skills Handbook page 757 and Lesson 1-8.) Simplify each absolute value. 1. |4 – 8| 2. |10 – (–5)| 3. |–2 – 6| 4. A(2, 3), B(5, 9) 5. K(–1, –3), L(0, 0) 6. W(4, –7), Z(10, –2) 7. C(–5, 2), D(–7, 6) 8. M(–1, –10), P(–12, –3) 9. Q(–8, –4), R(–3, –10) Find the distance between the points to the nearest tenth.

Finding perimeter and circumference. Objective 1 Finding area. Objective 2

Key Concepts The perimeter of a polygon is The area of a polygon is When finding the perimeter or area of a polygon be sure to use the same units for all the dimensions.

Example 1 Margaret’s garden is a square 12 ft on each side. Margaret wants a path 1 ft wide around the entire garden. What will the outside perimeter of the path be?

Example 2 G has a radius of 6.5 cm. Find the circumference of G in terms of . Then find the circumference to the nearest tenth.

Example 3 Quadrilateral ABCD has vertices A(0, 0), B(9, 12), C(11, 12), and D(2, 0). Find the perimeter.

Example 4 To make a project, you need a rectangular piece of fabric 36 in. wide and 4 ft long. How many square feet of fabric do you need?

Example 5 Find the area of B in terms of .

Key Concepts Postulate: If two figures are congruent, then their areas are equal. Postulate: The area of a region is the sum of the areas of its non-overlapping parts

Example 6 Find the area of the figure below.

Lesson Quiz 1. Find the perimeter in inches. 2. Find the area in square feet. 3. The diameter of a circle is 18 cm. Find the area in terms of . 4. Find the perimeter of a triangle whose vertices are X(–6, 2), Y(8, 2), and Z(3, 14). 5. Find the area of the figure below. All angles are right angles. A rectangle is 9 ft long and 40 in. wide.

Homework Pages 65 – 67 6 – 32 even, 37 – 39, 47 – 49, 59 – 62 Pages 71 – 73 Chapter Review Due Thursday Chapter 1 Test Friday, 9/23

Chapter 1: Tools of Geometry

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