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Chapter 3: Parallel and Perpendicular Lines

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Chapter 3: Parallel and Perpendicular Lines Parallel and Perpendicular Lines • Lessons 3-1, 3-2, and 3-5 Identify angle Key Vocabulary relationships that occur with parallel lines and a • parallel lines (p. 126) transversal, and identify and prove lines parallel • transversal (p. 127) from given angle relationships. • slope (p. 139) Use slope to analyze a line • Lessons 3-3 and 3-4 • equidistant (p. 160) and to write its equation. • Lesson 3-6 Find the distance between a point and a line and between two parallel lines. The framework of a wooden roller coaster is composed of millions of feet of intersecting lumber that often form parallel lines and transversals. Roller coaster designers, construction managers, and carpenters must know the relationships of angles created by parallel lines and their transversals to create a safe and stable ride. You will find how measures of angles are used in carpentry and construction in Lesson 3-2. 124 Chapter 3 Parallel and Perpendicular Lines Richard Cummins/CORBIS Prerequisite Skills To be successful in this chapter, you’ll need to master these skills and be able to apply them in problem-solving situations. Review these skills before beginning Chapter 3. For Lesson 3-1 Naming Segments Name all of the lines that contain the given point. (For review, see Lesson 1-1.) P Q R 1. Q 2. R 3. S 4. T S T For Lessons 3-2 and 3-5 Congruent Angles Name all angles congruent to the given angle. (For review, see Lesson 1-4.) 1 2 8 5. Є2 6. Є5 7 3 4 6 5 7. Є3 8. Є8 For Lessons 3-3 and 3-4 Equations of Lines For each equation, find the value of y for the given value of x. (For review, see pages 736 and 738.) 2 9. y ϭ 7x Ϫ 12, for x ϭ 3 10. y ϭϪᎏᎏx ϩ 4, for x ϭ 8 11. 2x Ϫ 4y ϭ 18, for x ϭ 6 3 Parallel and Perpendicular Lines Make this Foldable to help you organize information from this chapter. Begin with one sheet of 8ᎏ1ᎏ-inch by 11-inch paper. 2 Fold Fold Again Fold in half Unfold and fold matching the the long side short sides. up 2 inches to form a pocket. Staple or Glue Label Staple or glue the Label each side as outer edges to shown. Use index complete the cards to record pocket. examples. Parallel Perpendicular Reading and Writing As you read and study the chapter, write examples and notes about parallel and perpendicular lines on index cards. Place the cards in the appropriate pocket. Chapter 3 Parallel and Perpendicular Lines 125 Parallel Lines and Transversals • Identify the relationships between two lines or two planes. ᐉ • Name angles formed by a pair of lines and a m Vocabulary transversal. • parallel lines are parallel lines and planes • parallel planes used in architecture? • skew lines • transversal Architect Frank Lloyd Wright designed • consecutive interior many buildings using basic shapes, lines, angles and planes. His building at the right has • alternate exterior angles several examples of parallel lines, parallel • alternate interior angles planes, and skew lines. • corresponding angles RELATIONSHIPS BETWEEN LINES AND PLANES Lines ᐉ and m are coplanar because they lie in the same plane. If the lines were extended indefinitely, Study Tip they would not intersect. Coplanar lines that do not intersect are called parallel lines. Segments and rays contained within parallel lines are also parallel. Reading Math The term parallel and the ࿣ The symbol means is parallel to. Arrows are P Q notation ࿣ are used for used in diagrams to indicate that lines are parallel. ៭៮៬ lines, segments, rays, and In the figure, the arrows indicate that PQ is planes. The symbol ͞࿣ ៭៮៬ parallel to RS. RS means is not parallel to. Similarly, two planes can intersect or be parallel. PQϭ RS In the photograph above, the roofs of each level are contained in parallel planes. The walls and the floor of each level lie in intersecting planes. Draw a Rectangular Prism A rectangular prism can be drawn using parallel lines and parallel planes. Step 1 Draw two parallel planes Step 2 Draw the edges. Make Step 3 Label the vertices. to represent the top and any hidden edges of the bottom of the prism. prism dashed. DA C B H E G F Analyze 1. Identify the parallel planes in the figure. 2. Name the planes that intersect plane ABC and name their intersections. 3. Identify all segments parallel to BෆFෆ. 126 Chapter 3 Parallel and Perpendicular Lines Robert Holmes/CORBIS Notice that in the Geometry Activity, AෆEෆ and GෆFෆ do not intersect. These segments are not parallel since they do not lie in the same plane. Lines that do not intersect and are not coplanar are called skew lines. Segments and rays contained in skew lines are also skew. Example 1 Identify Relationships G a. Name all planes that are parallel to F plane ABG. plane CDE B A Study Tip b. Name all segments that intersect CෆHෆ. H E BෆCෆ, CෆDෆ, CෆEෆ, EෆHෆ, and GෆHෆ Identifying C Segments ෆෆ D Use the segments drawn c. Name all segments that are parallel to EF. in the figure even though AෆDෆ, BෆCෆ, and GෆHෆ other segments exist. d. Name all segments that are skew to BෆGෆ. AෆDෆ, CෆDෆ, CෆEෆ, EෆFෆ, and EෆHෆ ANGLE RELATIONSHIPS In the t transversal drawing of the railroad crossing, notice that the tracks, represented by line t, intersect the sides of the road, m represented by lines m and n. A line that intersects two or more lines in Study Tip a plane at different points is called a Transversals transversal. n The lines that the transversal intersects need not be parallel. Example 2 Identify Transversals AIRPORTS Some of the runways at O’Hare International Airport are shown below. Identify the sets of lines to which each given line is a transversal. a. line q ᐉ If the lines are extended, line q n intersects lines ᐉ, n, p, and r. p b. line m m lines ᐉ, n, p, and r c. line n Control Tower lines ᐉ, m, p, and q q d. line r lines ᐉ, m, p, and q r In the drawing of the railroad crossing above, notice that line t forms eight angles with lines m and n. These angles are given special names, as are specific pairings of these angles. www.geometryonline.com/extra_examples Lesson 3-1 Parallel Lines and Transversals 127 Transversals and Angles Name Angles Transversal p intersects Study Tip lines and . exterior angles Є1, Є2, Є7, Є8 q r Same Side Є Є Є Є p Interior Angles interior angles 3, 4, 5, 6 Consecutive interior consecutive interior angles Є3 and Є6, Є4 and Є5 1 2 angles are also called 4 3 q alternate exterior angles Є1 and Є7, Є2 and Є8 same side interior angles. Є Є Є Є 5 6 r alternate interior angles 3 and 5, 4 and 6 8 7 corresponding angles Є1 and Є5, Є2 and Є6, Є3 and Є7, Є4 and Є8 Example 3 Identify Angle Relationships Refer to the figure below. Identify each pair of angles as alternate interior, alternate exterior, corresponding, or consecutive interior angles. ᐉ a. Є1 and Є7b.Є2 and Є10 alternate exterior corresponding 1 2 a 4 3 c. Є8 and Є9d.Є3 and Є12 consecutive interior corresponding 5 6 8 7 b e. Є4 and Є10 f. Є6 and Є11 9 10 11 12 c alternate interior alternate exterior Concept Check 1. OPEN ENDED Draw a solid figure with parallel planes. Describe which parts of the figure are parallel. 2. FIND THE ERROR Juanita and Eric are naming alternate interior angles in the figure at the right. One of the angles must be Є4. Juanita Eric 2 1 3 Є4 and Є9 Є4 and Є10 4 Є4 and Є6 Є4 and Є5 5 6 9 10 8 7 12 Who is correct? Explain your reasoning. 11 3. Describe a real-life situation in which parallel lines seem to intersect. Guided Practice For Exercises 4–6, refer to the figure at the right. A B 4. Name all planes that intersect plane ADM. J K ෆෆ 5. Name all segments that are parallel to CD. C D 6. Name all segments that intersect KෆLෆ. M L 128 Chapter 3 Parallel and Perpendicular Lines Identify the pairs of lines to which p q each given line is a transversal. r 7. p 8. r 9. q 10. t t Identify each pair of angles as alternate a interior, alternate exterior, corresponding, 1 2 or consecutive interior angles. 4 3 6 Є Є 5 8 11. 7 and 10 7 12. Є1 and Є5 13. Є4 and Є6 9 10 b Є Є 11 12 14. 8 and 1 c Name the transversal that forms each pair of angles. Then identify the special name 1 2 5 6 ᐉ for the angle pair. 4 3 8 7 15. Є3 and Є10 9 10 13 14 m 12 11 16 15 16. Є2 and Є12 p q 17. Є 8 and Є 14 Application MONUMENTS For Exercises 18–21, refer to the photograph of the Lincoln Memorial. 18. Describe a pair of parallel lines found on the Lincoln Memorial. 19. Find an example of parallel planes. 20. Locate a pair of skew lines. 21. Identify a transversal passing through a pair of lines. Practice and Apply For Exercises 22–27, refer to the figure at the right. 22. Name all segments parallel to AෆBෆ. P Q For See Exercises Examples 23.
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