Parallel and Perpendicular Lines
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1 Chapter Chapter Parallel and Perpendicular Lines 3 Will the boats paths ever cross? If a sailboat heads directly into the wind, the sails flap and are useless. One solution is for boats to sail at an angle to the wind, as shown in the diagram wind wind During a race, sailboats often head into the wind at the same angle and they appear to be racing in parallel paths paths that never cross. Learn More About It You will learn more about paths of sailboats in Exercise 25 on p
2 Who uses Parallel and Perpendicular Lines? PHYSICAL THERAPIST Physical therapists use angles to measure the range of motion of elbows, knees, and other joints. These measurements help the therapist evaluate people s injuries. (p. 134) BICYCLE DESIGNER The angles in a bike frame influence the position of the rider. Georgena Terry builds bicycles whose designs feature a frame geometry more suitable for women. (p. 124) How will you use these ideas? Understand how escalators work. (p. 112) Learn how a compass and map are used in orienteering. (p. 119) See how surfing and kite flying are combined in the sport of kiteboarding. (p. 141) Understand how the strings of a guitar are related to the frets. (p. 148) 105
3 Chapter 3 Study Guide PREVIEW What th h pter b ut? Identifying relationships between lines Using properties of parallel and perpendicular lines Showing that two lines are parallel Key Words parallel lines, p. 108 perpendicular lines, p. 108 skew lines, p. 108 parallel planes, p. 109 transversal, p. 121 corresponding angles, p. 121 alternate interior angles, p. 121 alternate exterior angles, p. 121 same-side interior angles, p. 121 converse, p. 136 PREPARE Chapter Read s Q iz Take this quick quiz. If you are unsure of an answer, look at the reference pages for help. Vocabulary Check (refer to p. 75) 1. Name a pair of vertical angles. A a1 and a4 B a2 and a4 C a2 and a3 D a3 and a Skill Check (refer to pp. 75, 673) 2. The measure of a7 is 68, and a7 and a8 form a linear pair. What is the measure of a8? F 22 G 68 H 112 J Which of the following is a solution of the equation 5x 9 6x 11? A B 2 C 11 D 20 VISUAL STRATEGY Visualize It! Reading and Drawing Diagrams Use dashed lines to show parts of a figure that are hidden behind other parts. The dashed line shows that line n is not in plane S. is not in plane S. l P m S 106 Chapter 3 Parallel and Perpendicular Lines
4 Activity 3.1 Lines in Space Question How are lines related in space? Materials pencil straightedge lined paper Explore 1 Use a straightedge to draw two identical rectangles. 2 Connect the corresponding corners of the rectangles. 3 Erase parts of hidden lines to form dashed lines. Visualize It! Shading your sketch will help make it look three-dimensional. Think About It Using your sketch from the steps above, label the corners as shown. Then extend AD &* and CG&*. 1. Will ^&( AD and CG ^&( ever intersect in space? Lines that intersect on the page do not necessarily intersect in space. 2. Determine whether the following pairs of lines will intersect in space. a. BC ^&( and AE ^&( c. CD ^&( and DH ^&( b. GH ^&( and DH ^&( d. AB ^&( and EH ^&( Activity 107
5 3.1 Relationships Between Lines Goal Identify relationships between lines. Key Words parallel lines perpendicular lines skew lines parallel planes line perpendicular to a plane Two lines are parallel lines if they lie in the same plane and do not intersect. On the building, lines r and s are parallel lines. You can write this as r s. Triangles ( ) are used to indicate that the lines are parallel. Two lines are perpendicular lines if they intersect to form a right angle. Lines s and t are perpendicular lines. You can write this as s t. r s t EXAMPLE 1 Id tify Parall l a d Perp d lar Li Determine whether the lines are parallel, perpendicular, or neither. a. n and m b. p and q c. n and p Solution a. Lines n and m are parallel. b. Lines p and q are neither parallel nor perpendicular. c. Lines n and p are perpendicular. p m n q Two lines are skew lines if they do not lie in the same plane. Skew lines never intersect. In the diagram, lines c and b are skew lines. A c b Lines that intersect on the page do not necessarily intersect in space, such as lines f and g. f B Visualize It! h g EXAMPLE 2 Identify Skew Lines f Determine whether the lines are skew. a. f and h b. f and g B h Solution a. Lines f and h are not skew lines because they intersect. b. Lines f and g are skew lines. g 108 Chapter 3 Parallel and Perpendicular Lines
6 Ide tify Relatio ip twe in Use the diagram. 1. Name a pair of parallel lines. 2. Name a pair of perpendicular lines. z y x 3. Name a pair of skew lines. Two planes are parallel planes if they do not intersect. A line perpendicular to a plane is a line that intersects a plane in a point and that is perpendicular to every line in the plane that intersects it. t p A q F G Plane F is parallel to plane G. Line t is perpendicular to plane A. EXAMPLE 3 Id tify R lati h p Spa a. Name a plane that appears parallel to plane B. b. Name a line that appears perpendicular to plane B. D l m B Solution a. Plane C appears parallel to plane B. b. Line l appears perpendicular to plane B. n C Ide tify Relatio ip in Sp ce Think of each segment in the diagram as part of a line. 4. Name a line that is skew to VW ^&*(. R V 5. Name a plane that appears parallel to plane VWX. 6. Name a line that is perpendicular to plane VWX. P T S U X W 3.1 Relationships Between Lines 109
7 3.1 Exercises Guided Practice Vocabulary Check Skill Check 1. How are skew lines and parallel lines alike? How are they different? Fill in the blank with or to make the statement true. 2. Line k? line m. l m 3. Line m? line l. 4. Line l? line j. 5. Line k? line j. j k Match the photo with the corresponding description of the chopsticks. A. skew B. parallel C. intersecting Practice and Applications Extra Practice See p Line Relationships Determine whether the lines are parallel, perpendicular, or neither. 9. a and c 10. q and s 11. y and z a b q r x y c s z Homework H lp Example 1: Exs Example 2: Exs Example 3: Exs , Skew Lines Determine whether the lines are skew. Explain. 12. u and w 13. m and n 14. j and k u m j v n w l k 110 Chapter 3 Parallel and Perpendicular Lines
8 Identifying Relationships In Exercises 15 19, think of each segment in the diagram as part of a line. Fill in the blank with parallel, perpendicular, or skew. 15. DE ^&(, AB ^&(, and GC ^&( G appear to be?. 16. DE ^&( and BE ^&( are?. 17. BE ^&( and GC ^&( are?. 18. BE ^&( is? to plane DEF. A D H B E C F Balance 19. Plane GAD and plane CBE appear to be?. 20. Tightrope Walking Philippe Petit sometimes uses a long pole to help him balance on the tightrope. Are the rope and the pole at the left intersecting, perpendicular, parallel, or skew? PHILIPPE PETIT, the French highwire artist, walked more than 2000 feet up an inclined cable to the Eiffel Tower. The photo above is from a performance in New York City. Relationships in Space Think of each segment in the diagram below as part of a line. There may be more than one correct answer. 21. Name a line that appears parallel to QR ^&(. T 22. Name a line perpendicular to QR ^&(. X 23. Name a line skew to QR ^&(. 24. Name a plane that appears parallel to plane QRS. Visualize It! Sketch a figure that fits the description. 25. Three lines that are parallel 26. A line that is perpendicular to two parallel lines 27. Two planes that intersect 28. A line that is perpendicular to two parallel planes 29. A line that is perpendicular to two skew lines (Hint: Start by sketching a figure like the one above in Exercises ) P U R V S W Furniture Design In Exercises 30 33, use the photo of the chair shown at the right. 30. Name two pairs of parallel lines. h k 31. What kind of lines are h and m? 32. Name two lines that are skew. n j l 33. How many lines shown on the chair are perpendicular to j? m 3.1 Relationships Between Lines 111
9 Escalators In Exercises 34 and 35, use the following information. When a step on an up-escalator reaches the top, it flips over and plane A goes back down to the bottom. On each step, let plane A be the plane you stand on. 34. As each step moves around the escalator, is plane A always parallel to the ground level? Explain. 35. When a person is standing on plane A, is it parallel to ground level? Explain. ground level drive wheel EXAMPLE A prism is a three-dimensional figure with two identical faces, called bases, that lie in parallel planes as shown below. a. Sketch a prism with bases that have six sides. Label the prism. b. Name two edges that appear parallel. Solution a. To draw the prism, follow these steps. 1 Sketch a Prism Sketch two identical bases in parallel planes. In this case, the bases have six sides. 2 Prism base base Connect the bases and make hidden edges dashed. Label the prism. A F B E C D Student Help VISUAL STRATEGY Use dashed lines to show parts of a figure that are hidden behind other parts. G M H L J K b. Edges AG &* and DK & * appear parallel. Prisms Sketch a prism with bases that have the given number of sides. Label the prism. Name two edges that appear parallel. 36. Three sides 37. Four sides 38. Five sides 112 Chapter 3 Parallel and Perpendicular Lines
10 Challenge Fill in the blank with always, sometimes, or never. 39. Two skew lines are? parallel. 40. Two perpendicular lines? intersect. 41. Two skew lines are? coplanar. Standardized Test Practice 42. Multiple Choice Two lines are? lines if they do not lie in the same plane and they do not intersect. A perpendicular B parallel C coplanar D skew 43. Multiple Choice Use the diagram below to determine which of the following statements is false. Think of each segment in the diagram as part of a line. F G H J QP ^&( and MP ^&( are not parallel. MP ^&( and NR ^&( are skew. JM ^&( and KS ^&( are perpendicular. Plane KJM and plane QPM are not parallel. S P K N M P J R Mixed Review If-Then Statements Identify the hypothesis and the conclusion of the if-then statement. (Lesson 2.5) 44. If the band plays, then each member gets $ If ma5 120, then a5 is obtuse. 46. If there is a sale, then the store will be crowded. 47. If we can get tickets, then we ll go to the movies. Properties of Congruence Use the property to complete the statement. (Lesson 2.6) 48. Reflexive Property of Congruence:? caxyz 49. Symmetric Property of Congruence: If a1 ca2, then? c?. 50. Transitive Property of Congruence: If AB&* c EF&* and EF&* c ST&*, then? c?. Algebra Skills Reciprocals Find the reciprocal. (Skills Review, p. 656) Integers Evaluate. (Skills Review, p. 663) ( 3) ( 6) ( 5) ( 8)( 10) Relationships Between Lines 113
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