Unit 7: Parallel and Perpendicular Lines

1. Resource: Prentice Hall Unit 7: Parallel and Perpendicular Lines You will: Learn theorems about parallel and perpendicular lines

2. Lines are parallel when they are coplanar and they do not intersect A line that crosses two parallel lines is called a transversal. Text Resource: Prentice Hall 3.2: Properties of Parallel Lines

3. 4 3 3 3 8 7 7 7 2 2 2 1 6 6 6 5 Resource: Prentice Hall The ANGLES created by the intersection of a transversal with parallel lines have some interesting properties: There are several angle pairs created by this intersection. Corresponding Angles Alternate Interior Angles Same Side Interior Angles

4. 4, 8, and 5 are on the same side of the transversal as 1, but only 1 and 8 are interior. So 1 and 8 are same-side interior angles. Saint Agnes Academy Text Resource: Prentice Hall • Use the diagram above. • Identify which angle forms a pair of same-side interior angles with 1. • Identify which angle forms a pair of corresponding angles with 1. Same-side interior angles are on the same side of transversal t between lines p and q. 3-1

5. (continued) Corresponding angles also lie on the same side of the transversal. The angle corresponding to 1 must lie in the same position relative to line q as 1 lies relative to line p. Because 1 is an interior angle, 1 and 5 are corresponding angles. Saint Agnes Academy Text Resource: Prentice Hall GEOMETRY LESSON 3-1 Properties or Parallel Lines One angle must be an interior angle, and the other must be an exterior angle. 3-1

6. Compare 2 and the vertical angle of 1. Classify the angles as alternate interior angles, same–side interior angles, or corresponding angles. The vertical angle of 1 is between the parallel runway segments. 2 is between the runway segments and on the opposite side of the transversal runway. Because alternate interior angles are not adjacent and lie between the lines on opposite sides of the transversal, 2 and the vertical angle of 1 are alternate interior angles. Saint Agnes Academy Text Resource: Prentice Hall GEOMETRY LESSON 3-1 Properties or Parallel Lines x 3-1

7. a b Saint Agnes Academy Text Resource: Prentice Hall Corresponding Angle Postulate: If a transversal intersects two parallel lines, then corresponding angles are congruent. 2 1

8. a b Saint Agnes Academy Text Resource: Prentice Hall Use the Corresponding Angles Postulate to prove 1  3 2 Given: a || b Prove: 1  3 3 1 t 1. a || b 1. Given 2. 1  2 2. Corr. s Postulate 3. Vertical s Thm (2-1) 3. 3  2 4. Transitive Property (ch 2) 4. 1  3

9. a b Saint Agnes Academy Text Resource: Prentice Hall You have just proved the Alternate Interior Angles Theorem If a transversal intersects two parallel lines, then alternate interior angles are congruent. 3 1

10. a b Use the Alternate Interior Angles Theorem to prove m1 + m4 = 180 Given: a || b Prove: m1 + m4 = 180 3 4 1 t 1. a || b 1. Given 2. 1  3 2. Alt Int s Thm 3. Def of congruent 3. m1 = m 3 4. m3 + m 4 = 180 4.  Add’n Postulate (1-8) 5. m1+ m 4 = 180 5. Substitution

11. a b Saint Agnes Academy Text Resource: Prentice Hall You have just proved the Same Side Interior Angles Theorem If a transversal intersects two parallel lines, then same-side interior angles are supplementary. 4 1

12. 3 and 2 are adjacent angles that form a straight angle. m 3 + m 2 = 180 because of the Angle Addition Postulate. Saint Agnes Academy Text Resource: Prentice Hall Which theorem or postulate gives the reason that m 3 + m 2 = 180? 3-1

13. Saint Agnes Academy Text Resource: Prentice Hall In the diagram above, l || m. Find m 1 and then m 2. m 1 = 42 Corr s Postulate m 1 + m 2 = 180  Addition Postulate Substitution 42 + m 2 = 180 Subtraction Property of Equality. m 2 = 138 3-1

14. In the diagram above, || m. Find the values of a, b, and c. a = 65 Alternate Int s Thm c = 40 Alternate Int s Thm a + b + c = 180 Angle Add’n Postulate 65 + b + 40 = 180 Substitution b + 105= 180 Additon b = 75 Subtraction Prop of = 3-1

15. Saint Agnes Academy Text Resource: Prentice Hall Homework 3.1 page 118 Due at the beginning of the next class. Name Section # Page # Remember the honor code. No Copying! Show your work here IN PENCIL I pledge that I have neither given nor received aid on this assignment

16. Pages 118-121 Exercises 5. 1 and 2: corr. 3 and 4: alt. int. 5 and 6: corr. 6. 1 and 2: same-side int. 3 and 4: corr. 5 and 6: corr. 7. 1 and 2: corr. 3 and 4: same-side int. 5 and 6: alt. int. 8. alt. int. 9.a. 2 b. 1 c. corr. 10.a. Def. of b. Def. of right c. Corr. of i lines are . d. Subst. e. Def. of right f. Def. of 1.PQ and SR with transversal SQ; alt. int. 2.PS and QR with transversal SQ; alt. int. 3.PS and QR with transversal PQ; same-side int. 4.PS and QR with transversal SR; corr. s s s s s s s s s s s s s s s Saint Agnes Academy Text Resource: Prentice Hall Check in ink! GEOMETRY LESSON 3-1 3-1

17. 11. m 1 = 75 because corr. of || lines are ; m 2 = 105 because same-side int. of || lines are suppl. 12.m 1 = 120 because corr. of || lines are ; m 2 = 60 because same-side int. of || lines are suppl. 13.m 1 = 100 because same-side int. of || lines are suppl.; m 2 = 70 because alt. int. of || lines have = measure. 11. m 1 = 75 because corr. of || lines are ; m 2 = 105 because same-side int. of || lines are suppl. 12.m 1 = 120 because corr. of || lines are ; m 2 = 60 because same-side int. of || lines are suppl. 13.m 1 = 100 because same-side int. of || lines are suppl.; m 2 = 70 because alt. int. of || lines have = measure. 14. 70; 70, 110 15. 25; 65 16. 20; 100, 80 17.m 1 = m 3 = m 6 = m 8 = m 9 = m 11 = m 13 = m 15 = 52; m 2 = m 4 = m 5 = m 7 = m 10 = m 12 = m 14 = 128 18. You must find the measure of one . All that are vert., corr., or alt. int. to that will have that measure. All other will be the suppl. of that measure. 19. two 20. four 21. two 22. four 23. 32 s s s s s s s s s s s s s s Saint Agnes Academy Text Resource: Prentice Hall Check in ink! GEOMETRY LESSON 3-1 3-1

18. s s s s s s s s s Saint Agnes Academy Text Resource: Prentice Hall Check in ink! GEOMETRY LESSON 3-1 28. Answers may vary. Sample: E illustrates corr. ( 1 and 3, 2 and 4) and same-side int. ( 1 and 2, 3 and 4); I illustrates alt. int. ( 1 and 4, 2 and 3) and same-side int. ( 1 and 3, 2 and 4). 29. a. alt. int. b. He knew that alt. int. of || lines are . 30.a. 57 b. same-side int. 24.x = 76, y = 37, v = 42, w = 25 25.x = 135, y = 45 26. The labeled are corr. and should be . If you solve 2x – 60 = 60 – 2x, you get x = 30. This would be impossible since 2x – 60 and 60 – 2x would equal 0. 27.Trans means across or over. A transversal cuts across other lines. 3-1

19. s s s s Saint Agnes Academy Text Resource: Prentice Hall Check in ink! GEOMETRY LESSON 3-1 31.a. If two lines are || and cut by a transversal, then same-side ext. are suppl. b.Given: a || b Prove: 4 and 5 are suppl. 1.a || b (Given) 2.m 5 + m 6 = 180 ( Add. Post.) 3. 4 6 (Corr. are ) 4.m 5 + m 4 = 180 (Subst.) 5. 4 and 5 are suppl. (Def. of suppl.) 32.1.a || b (Given) 2. 1 2 (Vert. are .) 3. 2 3 (Corr. are .) 4. 1 3 (Trans. Prop.) 33. Never; the two planes do not intersect. 34. Sometimes; if they are ||. 35. Sometimes; they may be skew. 36. Sometimes; they may be ||. 37. D 38. G 39. D 40. I 3-1

20. 41.[2] a.  First show that 1 7. Then show that 7 5. Finally, show that 1 5 (OR other valid solution plan). b.  1 7 because vert. are . 7 5 because corr. of || lines are . Finally, by the Trans. Prop. of , 1 5. [1] incorrect sequence of steps OR incorrect logical argument s s Saint Agnes Academy Text Resource: Prentice Hall Check in ink! GEOMETRY LESSON 3-1 42. 121 43. 59 44. 29.5 45. (0.5, 7) 46. (–0.5, 3.5) 47. (3, 3) 48. add 4; 20, 24 49. multiply by –2; 16, –32 50. subtract 7; –5, –12 3-1

21. 1. Complete: and 4 are alternate interior angles. 2. Complete: and 8 are corresponding angles. 3. Suppose that m 3 = 37. Find m 6. 4. Suppose that m 1 = x + 12 and m 5 = 3x – 36. Find x. 5. If a transversal intersects two parallel lines, then same-side exterior angles are supplementary. Write a Plan for Proof. Given: m || n Prove: 2 and 7 are supplementary. 6 4 Show that m 2 = m 6. Then show that m 6 + m 7 = 180, and substitute m 2 for m 6. Saint Agnes Academy Text Resource: Prentice Hall GEOMETRY LESSON 3-1 Extra Practice In the diagram below, m || n. Use the diagram for Exercises 1–5. 143 24 3-1