1. CONFIDENTIAL 1 Geometry Proving Lines Parallel

2. CONFIDENTIAL 2 Warm Up Identify each of the following: 1) One pair of parallel segments 2) One pair of skew segments 3) One pair of perpendicular segments A B C D E

3. CONFIDENTIAL 3 Proving lines Parallel The converse of a theorem is found by exchanging the hypothesis and conclusion. The converse of a theorem is not automatically true. If it is true, it must be stated as postulate or proved as a separate theorem.

4. CONFIDENTIAL 4 POSTULATE: If two coplanar lines are cut by a transversal so that a pair of corresponding angles are congruent, then the two lines are parallel. Converse of Corresponding angle postulate 1 2 1 2 m n HYPOTHESIS: CONCLUSION: m || n

5. CONFIDENTIAL 5 Using the Converse of Corresponding angle postulate Use the converse of Corresponding Angles postulate and the given information to show that l || m. 1 2 l m 3 4 5 6 7 8 /1 = /5 /1 = /5 are Corresponding Angles. l || m Converse of Corr. /s Angles postulate. A) /1 = /5

6. CONFIDENTIAL 6 B) m/4 = (2x + 10) 0, m/8 = (3x - 55) 0, x = 65 1 2 l m 3 4 5 6 7 8 m/4 = 2(65) + 10 = 140 Substitute 65 for x. m/8 = 3(65) – 55 = 140 Substitute 65 for x. m/4 = m/8 Trans. prop. of equality /4 = /8 Def. of cong. angles. l || m Converse of Corr. /s Angles postulate.

7. CONFIDENTIAL 7 Use the converse of Corresponding Angles postulate and the given information to show that p || q. Now you try! 1 2 3 4 5 6 7 8 p q t 1a) m/1 = m/3 1b) m/7 = (4x + 25) 0, m/5 = (5x + 12) 0, x = 13

8. CONFIDENTIAL 8 Through a point P not on line l, there is exactly one line parallel to l. Parallel postulate The converse of Corresponding Angles postulate is used to construct parallel lines. The parallel postulate guarantees that for any line l, you can always construct through a point that is not on l.

9. CONFIDENTIAL 9 Construction of Parallel lines Draw a line l and a point P not on l. Draw a line m through P that intersects l. Label the angle 1. l P l P l m 1 Construct an angle congruent to /1 at P. By the converse of corresponding angle postulate, l || m. STEP1: STEP2: STEP3: l P l m 1 2 n

10. CONFIDENTIAL 10 THEOREM: If two coplanar lines are cut by a transversal so that a pair of Alternate interior angles are congruent, then the two lines are parallel. Converse of Alternate interior angles theorem 1 2 m n HYPOTHESIS: CONCLUSION: m || n Proving lines parallel

11. CONFIDENTIAL 11 THEOREM: If two coplanar lines are cut by a transversal so that a pair of Alternate interior angles are congruent, then the two lines are parallel. Converse of Alternate exterior angles theorem 3 4 3 4 m n HYPOTHESIS: CONCLUSION: m || n

12. CONFIDENTIAL 12 THEOREM: If two coplanar lines are cut by a transversal so that a pair of Same side interior angles are supplementary, then the two lines are parallel. Converse of Same side interior angles theorem 5 6 m n HYPOTHESIS: CONCLUSION: m || n m/5 = m/6 = 180 0

13. CONFIDENTIAL 13 Converse of Alternate exterior angles theorem Proof 1 2 m n 3 Given: /1 = /2 Prove: l || m Proof: It is given that /1 = /2. Vertical angles are congruent, so /1 = /3. By the Transitive property of Congruence, /2 = /3. So, l || m by the Converse of Corresponding Angle Postulate.

14. CONFIDENTIAL 14 1 2 3 4 8 7 6 5 r s t Determining whether lines are parallel Use the given information and the theorems you have learnt to show that r || s. /2 = /6 /2 = /6 are Alternate interior Angles. r || s Converse of Alt. int. Angles theorem. A) /2 = /6

15. CONFIDENTIAL 15 B) m/6 = (6x + 18) 0, m/7 = (9x + 12) 0, x = 10 m/6 = 6(10) + 18 = 78 Substitute 10 for x. m/7 = 9(65) + 12 = 102 Substitute 10 for x. m/6 + m/7 = 78 + 12 = 180 /6 and /7 are same-side interior angles. l || m Converse of same-side interior angles theorem. 1 2 3 4 8 7 6 5 r s t

16. CONFIDENTIAL 16 Proving lines parallel Use the given information and the theorems you have learnt to show that r || s. r || s Given: l || m, /1 = /3 Prove: r || p Proof: l mm p r 1 2 3 StatementsReasons 1. l || m1. Given 2. /1 = /22. corr. Angles Post. 3. /1 = /33. Given 4. /2 = /34. trans. prop. Of congruency 5. r || p5. conv. Of Alt ext angles thm.

17. CONFIDENTIAL 17 Given: /1 = /4, /3 and /4 are supplementary. Prove: l || m Now you try! ll m n 1 2 3 4

18. CONFIDENTIAL 18 During a race, all members of a rowing team should keep the oars parallel on each side. If m/1 = (3x + 13) 0, m/2 = (5x - 5) 0, x = 9, show that the oars are parallel. Sports Application 1 2 A line through the center of the boat forms a transversal to the two oars on each side of the boat. /1 and /2 are corresponding angles. If /1 = /2, then the oars are parallel. m/6 = 3(9) + 13 = 40 m/7 = 5(9) - 5 = 40 Substitute 10 for x in each expression. m/1 = m/2, /1 = /2. The corresponding angles are congruent, so the oars are parallel by the converse of corresponding angles postulates.

19. CONFIDENTIAL 19 Now you try! 1 2 4) Suppose the corresponding angles on the opposite side of the boat measure (4y - 2) 0 and (3y + 6) 0, where y = 8. Show that the oars are parallel.

20. CONFIDENTIAL 20 Assessment Use the converse of Corresponding Angles postulate and the given information to show that p || q. 1 2 8 7 4 3 5 6 p q t 1) m/4 = m/5 2) m/1 = (4x + 16) 0, m/8 = (5x - 12) 0, x = 28 3) m/4 = (6x - 19) 0, m/5 = (3x + 14) 0, x = 11

21. CONFIDENTIAL 21 Use the given information and the theorems you have learnt to show that r || s. 1 2 3 8 7 6 5 4 r s 4) m/1 = m/5 5) m/3 + m/4 = 180 0 6) m/3 = m/7 7) m/4 = (13x - 4) 0, m/8 = (9x + 16) 0, x = 5 8) m/8 = (17x + 37) 0, m/7 = (9x - 13) 0, x = 6 9) m/2 = (25x + 7) 0, m/6 = (24x + 12) 0, x = 5

22. CONFIDENTIAL 22 10) Complete the following 2 column proof: XX Y V W 2 1 3 Given: /1 = /2, /3 = /1 Prove: XY || VW Proof: StatementsReasons 1. /1 = /2, /3 = /11. Given 2. /2 = /32. a._______ 3.b. ______3. c._______

23. CONFIDENTIAL 23 POSTULATE: If two coplanar lines are cut by a transversal so that a pair of corresponding angles are congruent, then the two lines are parallel. Converse of Corresponding angle postulate 1 2 1 2 m n HYPOTHESIS: CONCLUSION: m || n Let’s review

24. CONFIDENTIAL 24 Using the Converse of Corresponding angle postulate Use the converse of Corresponding Angles postulate and the given information to show that l || m. 1 2 l m 3 4 5 6 7 8 /1 = /5 /1 = /5 are Corresponding Angles. l || m Converse of Corr. /s Angles postulate. A) /1 = /5

25. CONFIDENTIAL 25 B) m/4 = (2x + 10) 0, m/8 = (3x - 55) 0, x = 65 1 2 l m 3 4 5 6 7 8 m/4 = 2(65) + 10 = 140 Substitute 65 for x. m/8 = 3(65) – 55 = 140 Substitute 65 for x. m/4 = m/8 Trans. prop. of equality /4 = /8 Def. of cong. angles. l || m Converse of Corr. /s Angles postulate.

26. CONFIDENTIAL 26 Through a point P not on line l, there is exactly one line parallel to l. Parallel postulate The converse of Corresponding Angles postulate is used to construct parallel lines. The parallel postulate guarantees that for any line l, you can always construct through a point that is not on l.

27. CONFIDENTIAL 27 Construction of Parallel lines Draw a line l and a point P not on l. Draw a line m through P that intersects l. Label the angle 1. l P l P l m 1 Construct an angle congruent to /1 at P. By the converse of corresponding angle postulate, l || m. STEP1: STEP2: STEP3: l P l m 1 2 n

28. CONFIDENTIAL 28 THEOREM: If two coplanar lines are cut by a transversal so that a pair of Alternate interior angles are congruent, then the two lines are parallel. Converse of Alternate interior angles theorem 1 2 m n HYPOTHESIS: CONCLUSION: m || n Proving lines parallel

29. CONFIDENTIAL 29 THEOREM: If two coplanar lines are cut by a transversal so that a pair of Alternate interior angles are congruent, then the two lines are parallel. Converse of Alternate exterior angles theorem 3 4 3 4 m n HYPOTHESIS: CONCLUSION: m || n

30. CONFIDENTIAL 30 THEOREM: If two coplanar lines are cut by a transversal so that a pair of Same side interior angles are supplementary, then the two lines are parallel. Converse of Same side interior angles theorem 5 6 m n HYPOTHESIS: CONCLUSION: m || n m/5 = m/6 = 180 0

31. CONFIDENTIAL 31 Converse of Alternate exterior angles theorem Proof 1 2 m n 3 Given: /1 = /2 Prove: l || m Proof: It is given that /1 = /2. Vertical angles are congruent, so /1 = /3. By the Transitive property of Congruence, /2 = /3. So, l || m by the Converse of Corresponding Angle Postulate.

32. CONFIDENTIAL 32 Proving lines parallel Use the given information and the theorems you have learnt to show that r || s. r || s Given: l || m, /1 = /3 Prove: r || p Proof: l mm p r 1 2 3 StatementsReasons 1. l || m1. Given 2. /1 = /22. corr. Angles Post. 3. /1 = /33. Given 4. /2 = /34. trans. prop. Of congruency 5. r || p5. conv. Of Alt ext angles thm.

33. CONFIDENTIAL 33 You did a great job today!