# Geometric Relationships

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1 Geometric Relationships 8 Lines and ngles LB Building Blocks of Geometry Measuring and Classifying ngles ngle Relationships Classifying Lines Parallel Line Relationships 8B Polygons LB 8-5 LB Classify Triangles Triangles ngles in Triangles Quadrilaterals Polygons Geometric Patterns 8C Polygon Relationships 8-2 re Name of Figu Pentagon Hexagon Heptagon Octagon des Number of Si Nonagon Decagon 10 Undecagon Dodecagon Congruence 8-10 Transformations LB Transformations in the Coordinate Plane 8-11 Line Symmetry LB Create Tessellations rtist KEYWORD: MR7 Ch8 412 rtists help us to see our world in new ways. They use their creativity in many different kinds of careers. rtists might design graphics for Web sites, draw cartoons, design textiles and furniture, paint murals, or even illustrate courtroom scenes. rtists work with many materials, such as different kinds of paints, paper, stone, metal, stained glass, and tile. The table shows some geometric figures that an artist might use in a design. Chapter 8 2 2N D P R IN T

2 Vocabulary Choose the best term from the list to complete each sentence. 1. closed figure with three sides is a?, and a closed figure with four sides is a?. 2.? is used to measure and draw angles clockwise counterclockwise horizontal protractor quadrilateral ruler triangle vertical The arrow inside the line that extends left circle is moving?. to right is?. Complete these exercises to review skills you will need for this chapter. Graph Ordered Pairs Use the coordinate plane for problems 5 8. Write the ordered pair for each point B 7. C 8. D Identify Polygons Tell how many sides and angles each figure has O B C D Identify Congruent Figures Which two figures are exactly the same size and shape but in different positions? 12. B C D Geometric Relationships 413

3 Study Guide: Preview Previously, you defined geometric shapes. identified congruent and similar figures. located points on a coordinate plane. You will study measuring angles. using angle measurements to classify angles as acute, obtuse, or right. identifying relationships involving angles in triangles and quadrilaterals. using congruence and similarity to solve problems. transforming figures on the coordinate plane and describing the transformation. You can use the skills learned in this chapter to solve problems and create geometric proofs by using angle and line relationships in geometry. to use transformations to create patterns in art class. Key Vocabulary/Vocabulario angle congruent line symmetry parallel lines perpendicular lines polygon quadrilateral rotation transformation vertex ángulo congruentes simetría axial líneas paralelas Vocabulary Connections líneas perpendiculares polígono cuadrilátero rotación transformación vértice To become familiar with some of the vocabulary terms in the chapter, consider the following. You may refer to the chapter, the glossary, or a dictionary if you like. 1. Congruent comes from the Latin word congruere meaning to agree, correspond. If two figures are congruent, do you think they look the same or different? 2. Parallel comes from the Greek words para meaning alongside and allenon meaning one another. If two lines are parallel where do you think they are located in relation to each other? 3. Polygon comes from the Greek words polus meaning many and gonia meaning angle. ccording to this information, what do you think a shape called a polygon includes? 4. Quadrilateral comes from the Latin words quadri meaning four and latus meaning sides. How many sides do you think a quadrilateral has? 414 Chapter 8

5 8-1 Building Blocks of Geometry Learn to describe figures by using the terms of geometry. Vocabulary point line plane line segment ray Georgia Performance Standards M6P3.d Use the language of mathematics to express mathematical ideas precisely. lso, M6P4.a, M6P4.b. The building blocks of geometry are points, lines, and planes. point is an exact location. P point P, P point is named by a capital letter. line is a straight path that line B, B, B extends without end in line B, B opposite directions. line is named by two points on the line. plane is a flat surface that plane LMN, extends without end in all M plane MLN, L directions. plane NLM N plane is named by three points on the plane that are not on the same line. EXMPLE 1 Identifying Points, Lines, and Planes Use the diagram to name each geometric figure. three points, C, and D Five points are labeled: points, B, C, D, and E. C B E two lines B and BE You can also write B and EB. D a point shared by two lines point B Point B is a point on B and BE. a plane plane DC Use any three points in the plane that are not on the same line. Write the three points in any order. 416 Chapter 8 Geometric Relationships

6 Line segments and rays are parts of lines. Use points on a line to name line segments and rays. line segment is line segment Y XY, XY, made of two endpoints X line segment YX, YX and all the points between the endpoints. line segment is named by its endpoints. ray has one ray JK, JK endpoint. From the J K endpoint, the ray extends without end in one direction only. ray is named by its endpoint first followed by another point on the ray. EXMPLE 2 Identifying Line Segments and Rays Use the diagram to give a possible name to each figure. three different line segments TU, UV, and TV You can also write UT, VU, and VT. T U V three ways to name the line UT, VU, and VT You can also write TU, UV, and TV. six rays TU, TV, VT, VU, UV, and UT another name for ray TU TV T is still the endpoint. V is another point on the ray. Think and Discuss 1. Name the geometric figure suggested by each of the following: a page of a book; a dot (also called a pixel) on a computer screen; the path of a jet across the sky. 2. Explain how XY is different from XY. 3. Explain how B is different from B. GPS M6P2.c, M6P3.b, M6P3.d 8-1 Building Blocks of Geometry 417

7 8-1 Exercises GUIDED PRCTICE See Example 1 Use the diagram to name each geometric figure. 1. two points 2. a line 3. a point shared by two lines 4. a plane Georgia Performance Standards M6P1.a, M6P3.a, M6P3.c, M6P3.d L KEYWORD: MR7 8-1 KEYWORD: MR7 Parent N K J M See Example 2 Use the diagram to give a possible name to each figure. 5. two different ways to name the line 6. four rays 7. another name for C B C INDEPENDENT PRCTICE See Example 1 Use the diagram to name each geometric figure. 8. three points F 9. one line E H 10. a point shared by a line and a ray D G 11. a plane See Example 2 Use the diagram to give a possible name to each figure. 12. two different line segments 13. six rays 14. another name for YX W X Y Z Extra Practice See page 728. Extra Practice p. 728 PRCTICE ND PROBLEM SOLVING Use the diagram to find a name for each geometric figure described. 15. a point shared by three lines 16. two points on the same line 17. two rays 18. another name for D 19. two different names for the same line Draw each geometric figure. 20. RS 21. LM 23. XY on YX 22. B 24. JK and JH on the same line B D C F 418 Chapter 8 Geometric Relationships

8 Geography Mapmakers often include a legend on the maps they create. The legend explains what each symbol or location on the map represents. W 25. Name the geometric figure suggested by each part of the map. a. City Hall and Gordon Middle School b. Highway 80 c. the section of the road from the park to the post office d. the road from City Hall past the police station 26. student rides her bike from Gordon Middle School to City Hall. She then rides to the city park, first passing through the intersection near the police station and then passing by the school. List the segments on the map that represent her route. 27. Critical Thinking Name a line segment, a ray, and a line that include the same two locations on the map, but do not include the city park. 28. What s the Error? student described the road from Gordon Middle School to City Hall as VY. What was the student s error? 29. Write bout It Explain why the road from City Hall that goes past the police station suggests a ray named VX rather than a ray named XV. 30. Challenge What are all the possible names for the line suggested by IH-45? V W X Y Z V X HWY 80 Z MP LEGEND Y IH-45 City Hall Post Office Police Station Gordon Middle School City Park 31. Multiple Choice Which figure is NOT found in the diagram? Line C Line segment B Point D Ray F R M T 32. Gridded Response How many endpoints does a ray have? Find the value of k in each equation. (Lesson 2-8) 33. k k k k 7 Write each improper fraction as a mixed number. (Lesson 4-6) Building Blocks of Geometry 419

9 8-2 Measuring and Classifying ngles Learn to name, measure, draw, and classify angles. Vocabulary angle vertex acute angle right angle obtuse angle straight angle EXMPLE 1 m XYZ is read the measure of angle XYZ. You can adjust the angle that a treadmill makes with the ground in order to have an easier or more intense workout. n angle is formed by two rays with a common endpoint, called the vertex. n angle can be named by its vertex or by its vertex and a point from each ray. The middle point in the name must be the vertex. The angle of the treadmill can be called F, EFG, or GFE. ngles are measured in degrees. Use the symbol to show degrees. Measuring an ngle with a Protractor Use a protractor to measure the angle. Place the center point of the protractor on the vertex of the angle. Place the protractor so that ray YZ passes through the 0 mark. Using the scale that starts with 0 along ray YZ, read the measure where ray YX crosses. Y The measure of XYZ is 75. Write this as m XYZ G E X Z F EXMPLE 2 Georgia Performance Standards M6P3.d Use the language of mathematics to express mathematical ideas precisely. lso, M6P4.a, M6P4.b, M6P4.c. Drawing an ngle with a Protractor Use a protractor to draw an angle that measures 150. Draw a ray on a sheet of paper. Place the center point of the protractor on the endpoint of the ray. Place the protractor so that the ray passes through the 0 mark. Make a mark at 150 above the scale on the protractor. Draw a ray from the endpoint of the first ray through the mark at Chapter 8 Geometric Relationships

10 You can classify an angle by its measure. n acute angle measures less than 90. Right angles are usually marked with a symbol. right angle measures exactly 90. n obtuse angle measures more than 90 and less than 180. straight angle measures exactly 180. EXMPLE 3 Classifying ngles Classify each angle as acute, right, obtuse, or straight. The angle measures more than 90 and less than 180, so it is an obtuse angle. The angle measures less than 90, so it is an acute angle. EXMPLE 4 rchitectural pplication C D n architect designed this floor plan B for a five-sided room of a house. Family Room Classify, B, and D in the floor plan. E right angle The angle is marked as a right angle. B obtuse angle The angle measures more than 90 and less than 180. D acute angle The angle measures less than 90. Think and Discuss GPS M6P1.d, M6P3.b, M6P3.d 1. Explain how you know which point is the vertex of XYZ. 2. Give an example of a right angle in your classroom. 3. Tell what type of angle is suggested by each of the following. a. an open book lying flat b. the corner of a sheet of paper c. the point of a pencil d. the hands of a clock at 12: Measuring and Classifying ngles 421

11 8-2 See Example 1 Exercises GUIDED PRCTICE Georgia Performance Standards M6P3.a, M6P3.c, M6P5.a Use a protractor to measure each angle KEYWORD: MR7 8-2 KEYWORD: MR7 Parent See Example 2 See Example 3 Use a protractor to draw an angle with each given measure Classify each angle as acute, right, obtuse, or straight See Example Kendra is planning a flower bed for her garden, which is shown in the figure. Classify each angle of the flower bed. G H J INDEPENDENT PRCTICE L K See Example 1 Use a protractor to measure each angle See Example 2 See Example 3 Use a protractor to draw an angle with each given measure Classify each angle as acute, right, obtuse, or straight See Example The figure shows the shape of a ceramic tile. Classify each of the tile s angles. F B C E D 422 Chapter 8 Geometric Relationships

12 Extra Practice See page 728. Extra Practice p. 728 PRCTICE ND PROBLEM SOLVING Use a protractor to draw each angle. 24. an acute angle whose measure is less than an obtuse angle whose measure is between 100 and a right angle History Classify the smallest angle formed by the hands on each clock The clock tower in Rome, Georgia, was originally built as a water tower in The clock, built in Waltham, Massachusetts, was added to the top a year later. 30. Critical Thinking Can two acute angles that share a vertex form a right angle? Justify your answer with a diagram. 31. What s the Error? student wrote that the measure of this angle is 156. Explain the error the student may have made, and give the correct measure of the angle. How can the student avoid making the same mistake again? 32. Write bout It Describe how an acute angle and an obtuse angle are different. 33. Challenge How many times during the day do the hands of a clock form a straight angle? 34. Multiple Choice The figure shows a plan for a skateboard ramp. What type of angle is B? cute B Right C Obtuse D Straight C B 35. Multiple Choice Which of the following is another name for PQR? F P G RQP H PRQ QPR J Write each decimal as a percent and a fraction. (Lesson 7-8) Find the percent of each number. (Lesson 7-9) % of % of % of % of Measuring and Classifying ngles 423

13 8-3 ngle Relationships Learn to understand relationships of angles. Vocabulary congruent vertical angles adjacent angles complementary angles supplementary angles ngle relationships play an important role in many sports and games. Miniature-golf players must understand angles to know where to aim the ball. In the miniature-golf hole shown, m 1 m 2, m 3 m 4, and m 5 m 6. When angles have the same measure, they are said to be congruent. P M N R 160 Q Georgia Performance Standards M63 Evaluate algebraic expressions, including those with exponents, and solve simple one-step equations using each of the four basic operations. lso, M6P1.b, M6P3.d. Vertical angles are formed opposite each other when two lines intersect. Vertical angles have the same measure, so they are always congruent. MRP and NRQ are vertical angles. MRN and PRQ are vertical angles. djacent angles are side by side and have a common vertex and ray. djacent angles may or may not be congruent. MRN and NRQ are adjacent angles. They share vertex R and RN. NRQ and QRP are adjacent angles. They share vertex R and RQ. EXMPLE 1 Identifying Types of ngle Pairs Identify the type of each angle pair shown and 2 are opposite each other and are formed by two intersecting lines. They are vertical angles and 4 are side by side and have a common vertex and ray. They are adjacent angles. 424 Chapter 8 Geometric Relationships

14 Complementary angles are two angles whose measures have a sum of LMN and NMP are complementary. L M N P Supplementary angles are two angles whose measures have a sum of 180. K GHK and KHJ are supplementary. G H J EXMPLE 2 Identifying an Unknown ngle Measure Find each unknown angle measure. The angles are complementary. 55 a 90 The sum of the measures is 90. a 35 a 55 The angles are supplementary. 75 b 180 The sum of the measures is 180. b 105 b 75 The angles are vertical angles. c 51 Vertical angles are congruent. c 51 ngles JGF and KGH are congruent. d e The sum of the measures is 180. d e 44 Each angle d 22 and e 22 measures half of 44. J F d 136 G e H K Think and Discuss GPS M6P1.d, M6P3.b 1. Give the measure of 2 if 1 and 2 are vertical angles and m Give the measure of 3 if 3 and 4 are supplementary and m Tell whether the angles in Example 1B are supplementary or complementary. 8-3 ngle Relationships 425

15 8-3 See Example 1 Exercises GUIDED PRCTICE Georgia Performance Standards M6P1.b, M6P3.a, M6P5.b Identify the type of each angle pair shown KEYWORD: MR7 8-3 KEYWORD: MR7 Parent See Example 2 Find each unknown angle measure. 3. The angles are complementary. 4. The angles are supplementary. a b See Example 1 INDEPENDENT PRCTICE Identify the type of each angle pair shown See Example 2 Find each unknown angle measure. 7. The angles are vertical angles. 8. The angles are supplementary. c 62 d 78 Extra Practice See page 728. Extra Practice p. 728 PRCTICE ND PROBLEM SOLVING Use the figure for Exercises Which angles are not adjacent to 3? 10. Name all the pairs of vertical angles that include If the m 6 is 72, what are the measures of 5, 7, and 8? 12. What is the sum of the measures of 1, 2, 3, and 4? Chapter 8 Geometric Relationships

16 Use the figure for Exercises Find the measure of VYW. 14. Find the measure of XYZ. W X Multi-Step Use the measures of VYW and XYZ to find the measure of WYX. V Y Z Find the measure of the angle that is complementary to each given angle. Use a protractor to draw both angles Find the measure of the angle that is supplementary to each given angle. Use a protractor to draw both angles ngles and B are complementary. If the measure of angle equals the measure of angle B, what is the measure of each angle? 25. ngles C and D are each complementary to angle F. How are angle C and angle D related? 26. Write a Problem Draw a pair of adjacent supplementary angles. Write a problem in which the measure of one of the angles must be found. 27. Write bout It Two angles are supplementary to the same angle. Explain the relationship between the measures of these angles. 28. Challenge The measure of angle is 38. ngle B is complementary to angle. ngle C is supplementary to angle B. What is the measure of angle C? 29. Multiple Choice Which type of angles are always congruent? djacent B Complementary C Supplementary D Vertical 30. Multiple Choice ngle J and angle K are supplementary. What is the measure of K if the measure of J is 75? F 15 G 25 H Find the missing value in each proportion. (Lesson 7-3) 31. n m p s Classify each angle as acute, right, obtuse, or straight. (Lesson 8-2) J 8-3 ngle Relationships 427

17 8-4 Classifying Lines Learn to classify the different types of lines. Vocabulary parallel lines perpendicular lines skew lines Georgia Performance Standards M6P4.c Recognize and apply mathematics in contexts outside of mathematics. lso, M6P1.b, M6P3.d. The photograph of the houses and the table below show some of the ways that lines can relate to each other. The yellow lines are intersecting. The purple lines are parallel. The green lines are perpendicular. The white lines are skew. Intersecting lines are lines that cross at one common point. W Z X Y Line YZ intersects line WX. YZ intersects WX. The red arrows on the lines show that the lines are parallel. Parallel lines are lines in the same plane that never intersect. B L Line B is parallel to line ML. B ML M Perpendicular lines intersect to form 90 angles, or right angles. T R S U Line RS is perpendicular to line TU. RS TU Skew lines are lines that lie in different planes. They are neither parallel nor intersecting. M L B Line B and line ML are skew. B and ML are skew. 428 Chapter 8 Geometric Relationships

18 EXMPLE 1 Classifying Pairs of Lines Classify each pair of lines. The lines are in the same plane. They do not appear to intersect. They are parallel. The lines cross at one common point. They are intersecting. The lines intersect to form right angles. They are perpendicular. The lines are in different planes and are not parallel or intersecting. They are skew. EXMPLE 2 Physical Science pplication The particles in a transverse wave move up and down as the wave travels to the right. What type of line relationship does this represent? Points along the ribbon move up and down. This wave moves to the right. The direction that the particles move forms a right angle with the direction that the wave is traveling. The lines are perpendicular. Think and Discuss GPS M6P2.c, M6P3.b 1. Give an example of intersecting, parallel, perpendicular, and skew lines or line segments in your classroom. 2. Determine whether two lines must be parallel if they do not intersect. Explain. 8-4 Classifying Lines 429

19 8-4 Exercises See Example 1 GUIDED PRCTICE Classify each pair of lines. Georgia Performance Standards M6P3.a, M6P3.c, M6P5.b KEYWORD: MR7 8-4 KEYWORD: MR7 Parent See Example 2 3. Jamal dropped a fishing line from a pier, as shown in the drawing. What type of relationship is formed by the lines? See Example 1 INDEPENDENT PRCTICE Classify each pair of lines See Example 2 7. The drawing shows where an archaeologist found two fossils. What type of relationship is formed by the lines suggested by the fossils? Extra Practice See page 728. Extra Practice p. 728 PRCTICE ND PROBLEM SOLVING Describe each pair of lines as parallel, skew, intersecting, or perpendicular Capitol Street intersects 1st, 2nd, and 3rd venues, which are parallel to each other. West Street and East Street are perpendicular to 2nd venue. a. Draw a map showing the six streets. b. Suppose East and West Streets were perpendicular to Capitol Street rather than 2nd venue. Draw a map showing the streets. 430 Chapter 8 Geometric Relationships

20 The lines in the figure intersect to form a rectangular box. 12. Name all the lines that are parallel to D. B D C 13. Name all the lines that are perpendicular to FG. 14. Name a pair of lines that are skew. 15. Name all the lines that are not parallel to and do not intersect DH. E F H G Tell whether each statement is always, sometimes, or never true. 16. Intersecting lines are parallel. 17. Intersecting lines are perpendicular. 18. Perpendicular lines are intersecting. 19. Parallel lines are skew. 20. Critical Thinking Use parallel, perpendicular, skew, or a combination of these terms to describe the lines on a sheet of graph paper. Explain your answer. 21. What s the Error? student drew two lines and claimed that the lines were both parallel and intersecting. Explain the error. 22. Write bout It Explain the similarities and differences between perpendicular and intersecting lines. 23. Challenge Lines x, y, and z are in a plane. If lines x and y are parallel and line z intersects line x, does line z intersect line y? Explain. 24. Multiple Choice Which types of lines never intersect when they are in the same plane? Intersecting B Parallel C Perpendicular D Skew 25. Multiple Choice Main Street and Elm Street meet at a 90 angle. Which term best describes the streets? F Intersecting G Parallel H Perpendicular J Skew 26. Extended Response student draws two lines on the same plane. He claims the lines are skew lines. Is he correct? Explain. What are the possible line types that the student drew? Graph and label each point on a coordinate grid. (Lesson 6-6) 27. (3, 4) 28. B(1, 5) 29. C(7, 1) 30. D 8 1 2, E 2, Find the measure of the angle that is complementary to each given angle. (Lesson 8-3) Classifying Lines 431

21 Parallel Line Relationships 8-4 Use with Lesson 8-4 Parallel lines are in the same plane and never intersect. You can use a straightedge and protractor to draw parallel lines. ctivity 1 Draw a line on your paper. Label two points and B. B KEYWORD: MR7 Lab8 Georgia Performance Standards M6P2.a, M6P2.b, M6P5.a Use your protractor to measure and mark a 90 angle at each point. Draw rays with endpoints and B through the marks you made with the protractor. B Place the point of your compass on point, and draw an arc through the ray. Use the same compass opening to draw an arc through the ray at point B. Label the points of intersection X and Y. X B Y Now use your straightedge to draw a line through X and Y. Use the symbol for parallel lines to indicate that B is parallel to XY. B XY X Y B 432 Chapter 8 Geometric Relationships

22 When a pair of parallel lines is intersected by a third line, the angles formed have special relationships. 2 Draw a pair of parallel lines and a third line that intersects them. Label the angles 1 through 8, as shown. ngles inside the parallel lines are called interior angles. The interior angles here are angles 3, 4, 5, and 6. ngles outside the parallel lines are called exterior angles. The exterior angles here are angles 1, 2, 7, and Measure each angle, and write its measurement inside the angle. Shade angles with the same measure with the same color. The interior angles with the same measure are called alternate interior angles. They are angles 3 and 6 and angles 4 and 5. The exterior angles with the same measure are called alternate exterior angles. They are angles 1 and 8 and angles 2 and 7. ngles in the same position when the third line intersects the parallel lines are called corresponding angles. Think and Discuss 1. Name three pairs of corresponding angles. 2. Tell the relationship between the measure of interior angles and the measure of exterior angles. Try This Follow the steps to construct and label the diagram. 1. Draw a pair of parallel lines, and draw a third line intersecting them where one angle measures Label each angle on the diagram using the measure you know. 8-4 Hands-On Lab 433

23 Quiz for Lessons 8-1 Through Building Blocks of Geometry Use the diagram to name each geometric figure. 1. three points 2. two lines 3. a point shared by two lines 4. a plane 5. two different line segments 6. two different rays L N O M Ready to Go On? 8-2 Measuring and Classifying ngles Use a protractor to measure each angle. Then classify each angle as acute, right, obtuse, or straight The quarterback of a football team throws a long pass, and the angle the path of the ball makes with the ground is 30. Draw an angle with this measurement. 8-3 ngle Relationships 12. If two angles are supplementary and one angle measures 97, what is the measure of the other angle? Find each unknown angle measure b c a d Classifying Lines Classify each pair of lines Chapter 8 Geometric Relationships

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