Proving Lines Parallel 3-3 Proving Lines Parallel Warm Up Lesson Presentation Lesson Quiz Holt Geometry

Warm up intro: The converse of a theorem is found by exchanging the hypothesis (beginning of the sentence) and the conclusion (end of the sentence. The converse of a theorem is not automatically true. For example: If the sun is shining, I can see my shadow outside. The converse: If I can see my shadow outside, the sun is shining.

Warm Up State the converse of each statement. 1. If a = b, then a + c = b + c. 2. If mA + mB = 90°, then A and B are complementary. 3. If AB + BC = AC, then A, B, and C are collinear. If a + c = b + c, then a = b. If A and  B are complementary, then mA + mB =90°. If A, B, and C are collinear, then AB + BC = AC.

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

We don’t know because we have no idea if the lines are parallel What does x equal? 100 We don’t know because we have no idea if the lines are parallel x

Since the lines are parallel, x=100. What does x equal? 100 Since the lines are parallel, x=100. x

What do you know about the lines? 100 They are not parallel. 25

What do you know about the lines? 100 The lines must be parallel. 100

Make an “if…then” statement about each figure’s angles and lines. If two parallel lines are cut by a transversal, then the corresponding angles are congruent. Converse: If two lines are cut by a transversal so that the corresponding angles are congruent, then the lines are parallel. 100 100

Make an “if…then” statement about each figure’s angles and lines. If two parallel lines are cut by a transversal, then the alternate interior angles are congruent. Converse: If two lines are cut by a transversal so that the alternate interior angles are congruent, then the lines are parallel. 50 50

Make an “if…then” statement about each figure’s angles and lines. If two parallel lines are cut by a transversal, then the alternate exterior angles are congruent. Converse: If two lines are cut by a transversal so that the alternate exterior angles are congruent, then the lines are parallel. 60 60

Make an “if…then” statement about each figure’s angles and lines. If two parallel lines are cut by a transversal, then the consecutive interior angles are supplementary. Converse: If two lines are cut by a transversal so that the consecutive interior angles are supplementary, then the lines are parallel. 80 100

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 4  8 4 and 8 are corresponding angles. ℓ || m Conv. of Corr. s Post.

Use the Converse of the Corresponding Angles Postulate and the given information to show that ℓ || m. (aka plug in the value of x and see if it gives you a true statement.) m3 = (4x – 80)°, m7 = (3x – 50)°, x = 30 m3 = 4(30) – 80 = 40 Substitute 30 for x. m7 = 3(30) – 50 = 40 Substitute 30 for x. m3 = m7 Trans. Prop. of Equality 3  7 Def. of  s. ℓ || m Conv. of Corr. s Post.

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 + 26)°, m5 = (5x + 12)°, x = 13 m7 = 4(13) + 26 = 76 Substitute 13 for x. m5 = 5(13) + 12 = 77 Substitute 13 for x. m7 = m5 ℓ is not parallel to m

70 + 110 = 180 Check It Out! Example 1b Use the Converse of the Corresponding Angles Postulate and the given information to show that ℓ || m. m2 = (3x + 10)°, m3 = (5x + 10)°, x = 20 m2 = 3(20) + 10 = 70 Substitute 13 for x. m3 = 5(20) + 10 = 110 Substitute 13 for x. m2 + m3 = 180 70 + 110 = 180 ℓ || m

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 – 67)°, x = 5 m2 = 10x + 8 = 10(5) + 8 = 58 Substitute 5 for x. m3 = 25x – 67 = 25(5) – 3 = 58 Substitute 5 for x.

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 – 67)°, x = 5 m2 + m3 = 58° + 58° = 116° 2 and 3 are same-side interior angles. r is not parallel to s

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. P ℓ

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.

Example 4 Continued m1 = 8x + 20 = 8(15) + 20 = 140 Substitute 15 for x. m2 = 2x + 10 = 2(15) + 10 = 40 Substitute 15 for x. m1+m2 = 140 + 40 1 and 2 are supplementary. = 180 The same-side interior angles are supplementary, so pieces A and B are parallel by the Converse of the Same-Side Interior Angles Theorem.

Check It Out! Example 4 What if…? Suppose the corresponding angles on the opposite side of the boat measure (4y – 2)° and (3y + 6)°, where y = 8. Show that the oars are parallel. 4y – 2 = 4(8) – 2 = 30° 3y + 6 = 3(8) + 6 = 30° The angles are congruent, so the oars are || by the Conv. of the Corr. s Post.