30. Think About a Plan If the rowing crew at the right strokes in unison, the oars sweep out angles of equal measure. Explain why the oars on each side of the shell stay parallel.
    • What type of information do you need to prove lines parallel?
    • How do the positions of the angles of equal measure relate?

Algebra Determine the value of x for which r || s . r vertical linevertical line s .  Then find m ∠ 1 m angle 1  and m ∠ 2 . m angle 2 .

31. m ∠ 1 = 80 − x , m ∠ 2 = 90 − 2 x m angle , 1 equals 80 , minus x comma m angle . 2 equals 90 minus 2 . x
32. m ∠ 1 = 60 − 2 x , m ∠ 2 = 70 − 4 x m angle . 1 equals 60 minus 2 . x comma m angle . 2 equals 70 minus 4 . x
33. m ∠ 1 = 40 − 4 x , m ∠ 2 = 50 − 8 x m angle . 1 equals 40 minus 4 . x comma m angle . 2 equals 50 minus 8 . x
34. m ∠ 1 = 20 − 8 x , m ∠ 2 = 30 − 16 x m angle . 1 equals 20 minus 8 . x comma m angle . 2 equals 30 minus 16 . x

Use the diagram below for Exercises 35–41.

Open-Ended Use the given information. State another fact about one of the given angles that will guarantee two lines are parallel. Tell which lines will be parallel and why.

35. ∠ 1 ≅ ∠ 3 angle 1 approximately equal to angle 3
36. m ∠ 8 = 110 , m ∠ 9 = 70 m angle . 8 equals 110 comma . m angle , 9 equals 70
37. ∠ 5 ≅ ∠ 11 angle 5 approximately equal to angle 11
38. ∠ 11 angle 11  and ∠ 12 angle 12  are supplementary.
39. Reasoning If ∠ 1 ≅ ∠ 7 , angle 1 approximately equal to angle 7 comma  what theorem or postulate can you use to show that l || n ? l vertical linevertical line n question mark

Image Long Description

Write a flow proof.

40. Proof Given: l || n , ∠ 12 ≅ ∠ 8 l vertical linevertical line n comma angle 12 approximately equal to angle 8

    Prove: j || k j vertical linevertical line k

41. Proof Given: j || k , m ∠ 8 + m ∠ 9 = 180 j vertical linevertical line k comma m angle , 8 plus , m angle , 9 equals 180

    Prove: l || n l vertical linevertical line n

C Challenge

Which sides of quadrilateral PLAN must be parallel? Explain.

42. m ∠ P = 72 , m ∠ L = 108 , m ∠ A = 72 , m ∠ N = 108 m angle . p equals 72 comma . m angle . l equals 108 comma . m angle . eh equals 72 comma . m angle , n equals 108
43. m ∠ P = 59 , m ∠ L = 37 , m ∠ A = 143 , m ∠ N = 121 m angle . p equals 59 comma . m angle . l equals 37 comma . m angle . eh equals 143 comma . m angle , n equals 121
44. m ∠ P = 67 , m ∠ L = 120 , m ∠ A = 73 , m ∠ N = 100 m angle . p equals 67 comma . m angle . l equals 120 comma . m angle . eh equals 73 comma . m angle , n equals 100
45. m ∠ P = 56 , m ∠ L = 124 , m ∠ A = 124 , m ∠ N = 56 m angle . p equals 56 comma . m angle . l equals 124 comma . m angle . eh equals 124 comma . m angle , n equals 56
46. Proof Write a two-column proof to prove the following: If a transversal intersects two parallel lines, then the bisectors of two corresponding angles are parallel. (Hint: Start by drawing and marking a diagram.)

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