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Name———————————————————————— lesson 3.3 Date ————————————— Investigating Geometry Activity: The Transitive Property of Parallel Lines For use before the lesson “Prove Lines are Parallel” Materials: Question explore straightedge and tracing paper What is the transitive property of parallel lines? Investigating Parallel Lines STEP 1 Draw Parallel Lines Place your straightedge on your paper and trace both sides to draw parallel lines. Label the first line as “line 1” and the second line as “line 2.” STEP 2 Draw Third Line Place your straightedge on your paper so that the top edge is lined up with line 2. Now trace the other side of the straightedge to draw a line that is parallel to line 2. Label this line as “line 3.” Line 1 Line 2 1 2 3 4 5 6 Line 3 STEP 4 Label Angles Label the angles formed by the transversal and line 1 as shown. Now, label the angles formed by the transversal and line 3 as shown. STEP 5 Compare Angles Place your tracing paper on top of angles a, b, c, and d. Trace these angles. Slide the tracing paper down so that the traced angles are on top of angles e, f, g, and h. Compare the angles. draw conclusions a b c d Line 1 Line 2 e f g h Line 3 Use your observations to complete the following. 1. How do the corresponding angles b and f compare? Lesson 3.3 2. How do the alternate interior angles c and f compare? 3-32 3. How do the alternate exterior angles a and h compare? 4. You drew lines 1, 2, and 3 so that line 1 is parallel to line 2, and line 2 is parallel to line 3. However, is line 1 parallel to line 3? Explain your answer. 5. Complete the statement: If two lines are parallel to the same line, then they are ? to each other. Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved. STEP 3 Draw Transversal Use your straightedge to draw a nonperpendicular transversal to all three lines. Geometry Chapter Resource Book CS10_CC_G_MECR710761_C3L03IG.indd 32 4/27/11 2:45:35 AM Lesson 3.2 Use Parallel Lines and Transversals, continued B Study Guide 1. Using the Vertical Angles Congruence Theorem, m∠ 8 5 658. By the Corresponding Angles Postulate, m∠ 4 5 658. Because ∠ 8 and ∠ 6 are corresponding angles, by the Corresponding Angles Postulate, you know that m∠ 6 5 658. 2. Using the Vertical Angles Congruence Theorem, m∠ 3 5 1158. By the Corresponding Angles Postulate, m∠ 7 5 1158. Because ∠ 3 and ∠ 1 are corresponding angles, by the Corresponding Angles Postulate, you know that m∠ 1 5 1158. 3. 68 4. 25 5. 12 6. 10 7. 10 8. 5 9. 12 10. 16 Problem Solving Workshop: Using Alternative Methods Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved. 1. 1158; by the Alternate Exterior Angles Theorem 2. 308; by the Consecutive Interior Angles Theorem Challenge Practice 1. m∠ 1 5 428, m∠ 2 5 1388, m∠ 3 5 1388, m∠ 4 5 428, m∠ 5 5 1328, m∠ 6 5 488, m∠ 7 5 488, m∠ 8 5 1328, m∠ 9 5 908, m∠ 10 5 908, m∠ 11 5 908, m∠ 12 5 908, m∠ 13 5 1328, m∠ 14 5 488, m∠ 15 5 488, m∠ 16 5 1328, m∠ 17 5 428, m∠ 18 5 1388, m∠ 19 5 1388, m∠ 20 5 428 2. m∠ 1 5 358, m∠ 2 5 1458, m∠ 3 5 1118, m∠ 4 5 698, m∠ 5 5 1118, m∠ 6 5 698, m∠ 7 5 1458, m∠ 8 5 358, m∠ 9 5 698, m∠ 10 5 1118, m∠ 11 5 698, m∠ 12 5 1118, m∠ 13 5 768, m∠ 14 5 1048, m∠ 15 5 768, m∠ 16 5 1048, m∠ 17 5 1048, m∠ 18 5 768, m∠ 19 5 1048, m∠ 20 5 768 3. m∠ 1 5 1008, m∠ 2 5 808, m∠ 3 5 808, m∠ 4 5 1008, m∠ 5 5 1008, m∠ 6 5 568, m∠ 7 5 248, m∠ 8 5 248, m∠ 9 5 568, m∠ 10 5 1008, m∠ 11 5 1568, m∠ 12 5 248, m∠ 13 5 248, m∠ 14 5 1568, m∠ 15 5 1248, m∠ 16 5 568, m∠ 17 5 1248, m∠ 18 5 568, m∠ 19 5 1008, m∠ 20 5 808, m∠ 21 5 1008, C answers Theorem; Corresponding Angles Postulate; Transitive Property of Congruence 20. Given; Alternate Exterior Angles Theorem; Given; Corresponding Angles Postulate; Transitive Property of Congruence m∠ 22 5 808, m∠ 23 5 1568, m∠ 24 5 248, m∠ 25 5 248, m∠ 26 5 1568, m∠ 27 5 1008, m∠ 28 5 568, m∠ 29 5 248, m∠ 30 5 248, m∠ 31 5 568, m∠ 32 5 1008 4. Sample answer: A D Statements Reasons } } } } i DC , AD i BC 1. Given 1. AB 2. ∠ A and ∠ B and 2. Consecutive ∠ B and ∠ C Interior are supplementary. Angles Theorem 3. m∠ A 1 m∠ B 5 1808, 3. Definition of m∠ B 1 m∠ C 5 1808 supplementary angles 4. m∠ A 1 m∠ B 5 4. Substitution m∠ B 1 m∠ C Property of Equality 5. m∠ A 5 m∠ C 5. Subtraction Property of Equality 6. ∠ A > ∠ C 6. Definition of congruent angles 5. x 5 67; Draw a line through the angle x8 that is parallel to both m and n. Then using the Alternate Interior Angles Theorem and the definition of supplementary angles, you can determine that x 5 35 1 32 5 67. Lesson 3.3 Prove Lines are Parallel Teaching Guide Diagram 2; the lines appear to never intersect. 1. ∠12 and ∠14; ∠11 and ∠13; the angles in each pair have the same measure. 2. ∠4 and ∠6; ∠3 and ∠5; the angles in each pair have different measures. 3. If the measures of the alternate interior angles are equal, the lines are parallel. 4. ∠9 and ∠16; ∠10 and ∠15; the angles in each pair have the same measure. 5. ∠1 and ∠7; ∠2 and ∠8; the angles in each pair have different measures. 6. If the measures of the alternate exterior angles are equal, the lines are parallel. Investigating Geometry Activity 1. They are congruent. 2. They are congruent. 3. They are congruent. 4. Yes, line 1 is parallel to line 3, because corresponding angles are Geometry Chapter Resource Book A35 c ongruent, alternate interior angles are congruent, and alternate exterior angles are congruent. 5. If two lines are parallel to the same line, then they are parallel to each other. Practice Level A 1. yes; Corresponding Angles Converse 13. p i q 14. neither 15. Given; Alternate Exterior Angles Theorem; Definition of Congruent Angles; Given; Definition of Congruent Angles; Alternate Interior Angles Converse 16. Given; Alternate Interior Angles Theorem; Definition of Congruent Angles; Given; Substitution; Definition of Supplementary Angles; Consecutive Interior Angles Converse 2. yes; Alternate Interior Angles Converse Study Guide 3. yes; Alternate Exterior Angles Converse 1. 18 2. Yes; the angle that corresponds with 708 has a measure of 708 because it is a linear pair with the angle that measures 1108. 3. 11 4. 8 5. 12 4. yes; Corresponding Angles Converse 5. no 6. yes; Alternate Interior Angles Converse 7. 40 8. 30 9. 30 10. 14 11. 32 12. 95 13. C 14. m i n 15. p i q 16. p i q 17. neither 18. Given; Corresponding Angles Postulate; Given; Transitive Property of Congruence; Corresponding Angles Converse 19. Each lane is parallel to the one next to it, so l1 i l2, l2 i l3, and l3 i l4. Then l1 i l3 by the Transitive Property of Parallel Lines. By continuing this reasoning, l1 i l4. So, the first lane is parallel to the last lane. Practice Level B 1. yes; Consecutive Interior Angles Converse 2. yes; Alternate Interior Angles Converse 3. no 4. 40 5. 109 6.115 7. 22 8. 5 9. 80 10. congruent 11. supplementary 12. congruent 13. Each row is parallel to the one next to it, so r1 i r2, r2 i r3, and so on. Then r1 i r3 by the Transitive Property of Parallel Lines. By continuing this reasoning, r1 i r5. So, the first row is parallel to the last row. 14. Given 15. Corresponding Angles Postulate 16. Given 17. Transitive Property of Congruence 18. Alternate Exterior Angles Converse 19. Given 20. Alternate Interior Angles Theorem 21. Given 22. Transitive Property of Congruence 23. Alternate Interior Angles Converse 24. Corresponding Angles Converse Practice Level C 1. no 2. no 3. yes; Alternate Exterior Angles Converse 4. 16 5. 15 6. 24 7. 45 8. 23 A36 9. 42 10. m i n 11. neither 12. neither Geometry Chapter Resource Book Problem Solving Workshop: Mixed Problem Solving @##$ , @BF ##$ 1. a. Sample answer: CG @##$ c. Plane DCG b. Sample answer: CG 2. Yes. Since line m intersects line j, the angle formed that is 498 and the one below it are supplementary. So, the angle below it is 1318. Since line n intersects line j, the angle formed that is 1318 and the one above it are supplementary. So, the angle above it is 498. Since a pair of corresponding angles are congruent, the lines m and n are parallel. 3. 1108 4. Sketches will vary. There are 6 sides that could be a transversal for a pair of opposite sides. 5. a. no b. yes c. Line l on any step is always parallel to line l on any other step because the line is always going across the width of the escalator. Plane A on any step is not always parallel to plane A on any other step. Plane A on the step that is rising up the escalator is not parallel to plane A on the step coming down underneath it. d. 2; When each step is going from facing upward to facing downward and when each step is going from facing downward to facing upward Challenge Practice 1. a. ∠ VXW 5 (180 2 x)8 b. ∠ WXZ 5 (180 2 y)8 c. ∠ VXZ 5 (360 2 x 2 y)8 2. p i q by the Corresponding Angles Converse; q i r by the Consecutive Interior Angles Converse; p i r by the Transitive Property of Parallel Lines; s i t by the Alternate Exterior Angles Converse 3. x 5 6, y 5 9 Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved. answers Lesson 3.3 Prove Lines are Parallel, continued