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Parallel and Perpendicular Lines 3-3 3-3 1. Plan GO for Help What You’ll Learn Check Skills You’ll Need • To relate parallel and Complete each statement with always, sometimes, or never. perpendicular lines Lesson 1-4 1. Two lines in the same plane are 9 parallel. sometimes . . . And Why 1 To relate parallel and perpendicular lines Examples 2. Perpendicular lines 9 meet at right angles. always To show how to fit sides of a picture frame together, as in Example 1 Objectives 1 2 3. Two lines in intersecting planes are 9 perpendicular. sometimes Real-World Connection Using Theorem 3-11 4. Two lines in parallel planes are 9 perpendicular. never Math Background Theorems 3-9, 3-10 and 3-11 are generally easy for students to remember. However, the phrase in a plane often needs clarification. Draw two parallel lines on the board. At one point on one of the lines, hold a pencil or pointer perpendicular to the chalkboard. Students should recognize that the pointer is not perpendicular to both lines because the three lines are not coplanar. A similar demonstration can be done for Theorem 3-10 with a set of perpendicular lines and a pointer. Finally, using a pointer and two parallel lines one can demonstrate that parallel lines do not need to lie in the same plane. Key Concepts E L U The two diagrams suggest ways to draw parallel lines. You can draw them (a) parallel to a given line, or (b) perpendicular to a given line. Theorems 3-9 and 3-10 guarantee that the lines you draw are indeed parallel. R Relating Parallel and Perpendicular Lines R 1 Theorem 3-9 a If two lines are parallel to the same line, then they are parallel to each other. b c a6b Theorem 3-10 In a plane, if two lines are perpendicular to the same line, then they are parallel to each other. m6n t m More Math Background: p. 124C n Theorem 3-10 includes the phrase in a plane. On the other hand, Theorem 3-9 is true for any three such lines, whether they are coplanar (Exercise 3) or noncoplanar. Proof See p. 124E for a list of the resources that support this lesson. Proof of Theorem 3-10 Study what is given, what you are to prove, and the diagram. Then write a paragraph proof. r Given: r ' t, s ' t 1 s 2 PowerPoint t Bell Ringer Practice Check Skills You’ll Need Prove: r 6 s Proof: &1 and &2 are right angles by the definition of perpendicular, so they are congruent. Since corresponding angles are congruent, r 6 s. Lesson 3-3 Parallel and Perpendicular Lines Special Needs Lesson Planning and Resources Below Level L1 Have students draw lines on both sides of a stationary ruler. Then have students move the ruler so only one side is aligned with a line, and draw a third line. Discuss why these lines are all parallel. learning style: tactile 141 For intervention, direct students to: Solving Linear Equations Lesson 1-4: Example 2 Extra Skills, Word Problems, Proof Practice, Ch. 1 L2 Have students demonstrate Theorems 3-9 and 3-10 using spaghetti or other classroom objects. Have them use the corner of an index card to measure 90° angles. learning style: tactile 141 Guided Instruction 1 Math Tip Real-World EXAMPLE Connection Woodworking To make a frame for a painting, a miter box and a backsaw are used to cut the framing at 458 angles. Explain why cutting the framing at this angle ensures that opposite sides of the frame will be parallel. corners cut to form 45⬚ Two adjacent 458 angles form angles a 908 angle. The opposite sides of the frame are perpendicular to the same side. Thus the opposite sides are parallel because two lines perpendicular to a third line are parallel. Ask: When are all the angles formed by a transversal congruent? When the transversal is perpendicular to parallel lines. PowerPoint Additional Examples 1 Suppose that the top and bottom pieces of a picture frame are cut to make 60° angles with exterior sides of the frame. The two sides should be cut at what angle to ensure that opposite sides of the frame will be parallel? 30° Quick Check 1 Can you assemble the framing at the right into a frame with opposite sides parallel? Explain. Yes; 30° ± 60° ≠ 90° 2 Study what is given, what you are to prove, and the diagram. Then write a paragraph proof. 60⬚ 60⬚ 60⬚ 60⬚ 30⬚ 30⬚ Theorems 3-9 and 3-10 gave conditions by which you can conclude that lines are parallel. Theorem 3-11 provides a way for you to conclude that lines are perpendicular. You will prove Theorem 3-11 in Exercise 11. s a Key Concepts Theorem 3-11 n In a plane, if a line is perpendicular to one of two parallel lines, then it is also perpendicular to the other. b ᐉ m n'm c Proof Given: In a plane, a s, c s, and a 6 b Prove: c 6 b Proof: Lines a and v are both to line s, so a n c because two lines perpendicular to the same line are parallel. It is given that a n b. Therefore, c n b because two lines parallel to the same line are parallel to each other. Using Theorem 3-11 a c b Given: In a plane, a ' b, b ' c, and c ' d. Prove: a ' d d Proof: Lines a and c are both perpendicular to line b, so a 6 c because two lines perpendicular to the same line are parallel. It is given that c ' d . Therefore, a ' d because a line that is perpendicular to one of two parallel lines is perpendicular to the other (Theorem 3-11). Quick Check Closure 142 EXAMPLE Study what is given, what you are to prove, and the diagram. Then write a paragraph proof. Resources • Daily Notetaking Guide 3-3 L3 • Daily Notetaking Guide 3-3— L1 Adapted Instruction Name two methods this lesson gives to prove that two lines are parallel. Show that two lines are parallel to the same line or that in a plane two lines are perpendicular to the same line. 2 142 2 From what is given in Example 2, can you also conclude b 6 d? Explain. Yes; b n d by Thm. 3-10. A * ) * ) In the rectangular solid shown here, AC and BD * ) * ) D C are parallel. EC is perpendicular to AC , but it is * ) not perpendicular to BD . This is why Theorem 3-11 G states that the lines must be “in a plane.” E F B H Chapter 3 Parallel and Perpendicular Lines Advanced Learners English Language Learners ELL L4 After students have examined Theorem 3-9, ask: Is the relationship “is parallel to” reflexive, symmetric, and/or transitive? symmetric and transitive learning style: verbal For Example 1, show a miter box and a backsaw borrowed from the woodworking teacher, or find pictures from a tool catalog. Students need to understand that these tools can make 45° cuts. learning style: visual EXERCISES For more exercises, see Extra Skill, Word Problem, and Proof Practice. 2. Practice Practice and Problem Solving Assignment Guide A Practice by Example 1. A carpenter is building a trellis for vines to grow on. The completed trellis will have two D sets of overlapping diagonal slats of wood. a. What must be true of &1, &2, and &3 if slats A, B, and C must be parallel? l1 O l2 O l3 A b. The carpenter attaches slat D so that it is 1 perpendicular to slat A. Is slat D perpendicular to slats B and C? Justify your answer. See back of book. 2. Study what is given, what you are to prove, and the diagram. Then write a proof. See margin. b Example 1 (page 142) GO for Help Example 2 Proof (page 142) 1 A B 1-15 B 2 C 3 B Apply Your Skills c For a guide to solving Exercise 3, see p. 145. Test Prep Mixed Review 24-25 26-29 To check students’ understanding of key skills and concepts, go over Exercises 1, 2, 8, 12, 13. d a 3. Developing Proof Copy and complete this paragraph proof of Theorem 3-9 for three coplanar lines. GO for Help 14-23 Homework Quick Check Given: In a plane, a ' b, b ' c, and c 6 d. Prove: a 6 d C Challenge Exercises 3 Theorem 3-9 is also true when the three lines are noncoplanar, but that proof is beyond the scope of the course. If two lines are parallel to the same line, then they are parallel to each other. Tactile Learners Given: O 6 k and m 6 k Exercise 13 Provide a physical object such as a shoebox to help students explain. 1 Prove: O 6 m Proof: O 6 k means that &2 > &1 by the a. 9 Postulate. m 6 k means that b. 9 > c. 9 for the same reason. By the Transitive Property of Congruence, Converse of Corr. ' &2 > &3. By the d. 9 Postulate, O 6 m. 2 k corr. ' l1, l3 (any order) ᐉ 3 m GPS Guided Problem Solving L3 L4 Enrichment Each of the following statements describes a ladder. What can you conclude about the rungs, one side, or both sides of each ladder? Explain. 4-10. See margin. 4. The rungs are each perpendicular to one side. L2 Reteaching L1 Adapted Practice Practice Name 5. The rungs are parallel and the top rung is perpendicular to one side. Class Practice 3-3 L3 Date Parallel Lines and the Triangle Angle-Sum Theorem Find the value of each variable. 6. The sides are parallel. The rungs are perpendicular to one side. 1. 2. 3 yⴗ 65ⴗ xⴗ 7. The rungs are perpendicular to one side. The other side is perpendicular to the top rung. 30ⴗ 60ⴗ nⴗ 39ⴗ 75ⴗ 68ⴗ 4. 5. 93ⴗ 6. 61ⴗ 46ⴗ mⴗ 36ⴗ 79ⴗ 8. Each side is perpendicular to the top rung. 7. 8. 10ⴗ 55ⴗ aⴗ bⴗ cⴗ 9. Each rung is parallel to the top rung. 44ⴗ 9. 28ⴗ zⴗ xⴗ pⴗ xⴗ 25ⴗ 53ⴗ yⴗ wⴗ 56ⴗ 62ⴗ vⴗ tⴗ Find the measure of each numbered angle. 10. The rungs are perpendicular to one side. The sides are not parallel. 10. 11. 1 12. 2 32ⴗ Proof 11. Prove Theorem 3-11: In a plane, if a line is GPS perpendicular to one of two parallel lines, then it is also perpendicular to the other. See margin. a c Given: In a plane, a ' b, and b 6 c. Real-World Connection This ladder’s rungs are perpendicular to each side. Therefore, the rungs are parallel to each other. b Prove: a ' c Proof 2. a is perp. to b and c is perp. to b, so a is parallel to c because two lines perp. to the same line are parallel. d is parallel to c and a is parallel to c, so by Thm. 3-9, a is parallel to d. 13. Lesson 3-3 Parallel and Perpendicular Lines 4. The rungs are parallel to each other because they are all perpendicular to the same side. 5. All of the rungs are perpendicular to one side. The side is perp. to the top rung, and 143 because all of the rungs are parallel to each other, the side is perp. to all of the rungs. 6. The rungs are perpendicular to both sides. The rungs are perp. to one of two 14. 2 69.7ⴗ 3 15. 3 4 5 38ⴗ 31ⴗ 46ⴗ 126.8ⴗ 70ⴗ 72ⴗ 86ⴗ 120ⴗ 116ⴗ 1 2 16. The sides of a triangle are 10 cm, 8 cm, and 10 cm. Classify the triangle. 17. The angles of a triangle are 44°, 110°, and 26°. Classify the triangle. Use a protractor and a centimeter ruler to measure the angles and the sides of each triangle. Classify each triangle by its angles and sides. 18. 12. Prove: If a line is perpendicular to each of two other lines, all in one plane, then the two other lines are parallel. s r 2 Given: t ' r, t ' s 1 t Prove: r 6 s This is a restatement of Thm. 3-10. lesson quiz, PHSchool.com, Web Code: aua-0303 © Pearson Education, Inc. All rights reserved. 140ⴗ 19. 20. parallel lines, so they are perp. to both lines. 7. The rungs are parallel to each other because they are all perpendicular to one side. The sides are parallel because they are both perpendicular to one rung. 8. The sides are parallel because they are both perpendicular to one rung. 143 4. Assess & Reteach PowerPoint Lesson Quiz B A 13. Writing Theorem 3-10: In a plane, two lines perpendicular to the same line are parallel. Use the rectangular solid at the right to explain why the words in a plane are needed. See margin. D C H G E Complete the proof. C Challenge a b x y GO z nline Homework Help Visit: PHSchool.com Web Code: aue-0303 Given: In a plane, a y, b x, a 6 y, and y 6 z Prove: x 6 z Proof: Because 1. a y and a 6 b, line y is perpendicular to line b by Theorem 2. 3-11. Lines x and y are both perpendicular to line b, so 3. x n y because two lines perpendicular to the same line are 4. parallel. Therefore, x 6 z because two lines that are 5. parallel to the same line are parallel to each other. a, b, c, and d are distinct lines in the same plane. Exercises 14-21 show different combinations of relationships between a and b, b and c, and c and d. For each combination of the three relationships, how are a and d related? 14. a 6 b, b 6 c, c 6 d a n d 15. a 6 b, b 6 c, c ' d a ' d 16. a 6 b, b ' c, c 6 d a ' d 17. a ' b, b 6 c, c 6 d a ' d 18. a 6 b, b ' c, c ' d a n d 19. a ' b, b 6 c, c ' d a n d 20. a ' b, b ' c, c 6 d a n d 21. a ' b, b ' c, c ' d a ' d Critical Thinking The Reflexive, Symmetric, and Transitive Properties for Congruence (O) are listed on page 105. 22–23. See margin. 22. Write reflexive, symmetric, and transitive statements for “is parallel to” ( 6 ). State whether each statement is true or false and justify your answer. 23. Repeat Exercise 22 for “is perpendicular to” ( '). Test Prep Multiple Choice 24. In a plane, line e is parallel to line f, line f is parallel to line g, and line h is perpendicular to line e. Which of the following CANNOT be true? C A. e 6 g B. h 6 f C. g 6 h D. e 6 h * ) * ) Alternative Assessment Have students draw and label three parallel lines and a perpendicular transversal. Students identify one or two angles as right and two lines as parallel. They then write a proof statement that requires use of the theorems from the lesson. Students trade papers with a partner and write paragraph proofs. 25. In a plane, AB is parallel to CD . &ABC is a right angle. What can you conclude? G * ) * ) I. AB ' BC F. I only * ) * ) II. BC ' CD G. I and II H. II only GO for Help * ) Lesson 3-2 x 2 Algebra Determine the value of x for which a n b. 26. 53 27. a a b 46 124⬚ b Test Prep Resources Lesson 2-2 For additional practice with a variety of test item formats: • Standardized Test Prep, p. 193 • Test-Taking Strategies, p. 188 • Test-Taking Strategies with Transparencies 44⬚ (2x + 18)⬚ Each conditional statement below is true. Write its converse. If the converse is also true, combine the statements into a biconditional. 28. If x 5 7, then x2 5 49. 144 144 * ) III. BC ' AD J. I, II and III Mixed Review (3x ⫺ 2)⬚ 9. All of the rungs are parallel. All of the rungs are parallel to one rung, so they are all parallel to each other. F If x2 ≠ 49, then x ≠ 7. 29. If two lines in a plane do not meet, then the lines are parallel. If two lines in a plane are parallel, they do not meet. Two lines in a plane do not meet if and only if the lines are parallel. Chapter 3 Parallel and Perpendicular Lines 10. The rungs are parallel because they are all perpendicular to one side. 11. In the diagram, a b means the marked l is a rt. l. b n c means that the corres. l formed by a and c is a rt. l, so a c. 13. Answers may vary. Sample: In the diagram, AB BH and AB BD, but BH o BD. They intersect. 22. Reflexive: a n a; false; any line intersects itself. Symmetric: If a n b, then b n a; true; b and a are coplaner and do not intersect. Transitive: In general, if a n b, and b n c; then a n c;