Identify parallel, perpendicular, and skew lines.

1. Objectives Identify parallel, perpendicular, and skew lines. Identify the angles formed by two lines and a transversal.

2. Vocabulary parallel lines perpendicular lines skew lines parallel planes transversal corresponding angles alternate interior angles alternate exterior angles same-side interior angles

3. D

4. Helpful Hint Segments or rays are parallel, perpendicular, or skew if the lines that contain them are parallel, perpendicular, or skew.

5. Example 1: Identifying Types of Lines and Planes Identify each of the following. A. a pair of parallel segments B. a pair of skew segments C. a pair of perpendicular segments D. a pair of parallel planes

6. Check It Out! Example 1 Identify each of the following. a. a pair of parallel segments b. a pair of skew segments c. a pair of perpendicular segments d. a pair of parallel planes

8. Example 2: Classifying Pairs of Angles Give an example of each angle pair. A. corresponding angles B. alternate interior angles C. alternate exterior angles D. same-side interior angles

9. Check It Out! Example 2 Give an example of each angle pair. A. corresponding angles B. alternate interior angles C. alternate exterior angles D. same-side interior angles

10. Helpful Hint To determine which line is the transversal for a given angle pair, locate the line that connects the vertices.

11. Example 3: Identifying Angle Pairs and Transversals Identify the transversal and classify each angle pair. A. 1 and 3 B. 2 and 6 C. 4 and 6

12. Check It Out! Example 3 Identify the transversal and classify the angle pair 2 and 5 in the diagram.

13. Assignment • Pg 149 #14-17, 26-32

15. Example 1: Using the Corresponding Angles Postulate Find each angle measure. A. mECF B. mDCE

16. Check It Out! Example 1 Find mQRS.

17. Helpful Hint If a transversal is perpendicular to two parallel lines, all eight angles are congruent.

18. Example 2: Finding Angle Measures Find each angle measure. A. mEDG B. mBDG

19. Check It Out! Example 2 Find mABD.

20. Example 3: Music Application Find x and y in the diagram.

21. Check It Out! Example 3 Find the measures of the acute angles in the diagram.

22. Assignment • Pg 158 #6-25, 31

23. Objective Use the angles formed by a transversal to prove two lines are parallel.

24. Recall that 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 a postulate or proved as a separate theorem.

26. Example 1A: Using the Converse of the Corresponding Angles Postulate Use the Converse of the Corresponding Angles Postulate and the given information to show that ℓ || m. 4 8

27. Example 1B: Using the Converse of the Corresponding Angles Postulate Use the Converse of the Corresponding Angles Postulate and the given information to show that ℓ || m. m3 = (4x – 80)°, m7 = (3x – 50)°, x = 30

28. Check It Out! Example 1a Use the Converse of the Corresponding Angles Postulate and the given information to show that ℓ || m. m1 = m3

29. Check It Out! Example 1b Use the Converse of the Corresponding Angles Postulate and the given information to show that ℓ || m. m7 = (4x + 25)°, m5 = (5x + 12)°, x = 13

30. The Converse of the Corresponding Angles Postulate is used to construct parallel lines. The Parallel Postulate guarantees that for any line ℓ, you can always construct a parallel line through a point that is not on ℓ.

32. Example 2A: Determining Whether Lines are Parallel Use the given information and the theorems you have learned to show that r || s. 4 8

33. Example 2B: Determining Whether Lines are Parallel Use the given information and the theorems you have learned to show that r || s. m2 = (10x + 8)°, m3 = (25x – 3)°, x = 5

34. Example 2B Continued Use the given information and the theorems you have learned to show that r || s. m2 = (10x + 8)°, m3 = (25x – 3)°, x = 5

35. Check It Out! Example 2a Refer to the diagram. Use the given information and the theorems you have learned to show that r || s. m4 = m8

36. Check It Out! Example 2b Refer to the diagram. Use the given information and the theorems you have learned to show that r || s. m3 = 2x, m7 = (x + 50), x = 50

37. Example 3: Proving Lines Parallel Given:p || r , 1 3 Prove: ℓ || m

38. Example 3 Continued 1.p || r 2.3  2 3.1  3 4. Trans. Prop. of 

39. Check It Out! Example 3 Given: 1 4, 3 and 4 are supplementary. Prove: ℓ || m

40. Check It Out! Example 3 Continued 1.1  4 2. m1 = m4 3.Given 4. m3 + m4 = 180 4. Trans. Prop. of  5. m3 + m1 = 180 6. m2 = m3 7. Substitution 8. ℓ || m

41. Example 4: Carpentry Application A carpenter is creating a woodwork pattern and wants two long pieces to be parallel. m1= (8x + 20)° and m2 = (2x + 10)°. If x = 15, show that pieces A and B are parallel.

42. Example 4 Continued The same-side interior angles are supplementary, so pieces A and B are parallel by the Converse of the Same-Side Interior Angles Theorem.

43. Assignment • Pg 166 #1-11, 30-35

44. Vocabulary perpendicular bisector distance from a point to a line

45. The _________________of a segment is a line perpendicular to a segment at the segment’s midpoint. The shortest segment from a point to a line is perpendicular to the line. This fact is used to define the ________________________ as the length of the perpendicular segment from the point to the line.

46. A. Name the shortest segment from point A to BC. Example 1: Distance From a Point to a Line B. Write and solve an inequality for x.

47. A. Name the shortest segment from point A to BC. Check It Out! Example 1 B. Write and solve an inequality for x.

48. HYPOTHESIS CONCLUSION

49. Example 3: Carpentry Application A carpenter’s square forms a right angle. A carpenter places the square so that one side is parallel to an edge of a board, and then draws a line along the other side of the square. Then he slides the square to the right and draws a second line. Why must the two lines be parallel?