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3-3 3-5 Proving ProvingLines LinesParallel Parallel Warm Up Lesson Presentation Lesson Quiz Holt Geometry Holt Geometry 3-5 Proving Lines Parallel 3-3 Warm Up State the converse of each statement. 1. If a = b, then a + c = b + c. If a + c = b + c, then a = b. 2. If mA + mB = complementary. If A and B are then mA + mB 3. If AB + BC = AC, 90°, then A and B are complementary, =90°. then A, B, and C are collinear. If A, B, and C are collinear, then AB + BC = AC. Holt Geometry 3-5 Proving Lines Parallel 3-3 Objective Use the angles formed by a transversal to prove two lines are parallel. Holt Geometry 3-3 3-5 Proving Lines Parallel What does x equal? 100 x Holt Geometry We don’t know because we have no idea if the lines are parallel 3-5 Proving Lines Parallel 3-3 What does x equal? 100 x Holt Geometry Since the lines are parallel, x=100. 3-5 Proving Lines Parallel 3-3 100 What do you know about the lines? They are not parallel. 25 Holt Geometry 3-5 Proving Lines Parallel 3-3 100 100 Holt Geometry What do you know about the lines? The lines must be parallel. 3-5 Proving Lines Parallel 3-3 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. 100 100 Holt Geometry Corresponding angles Converse: If two lines are cut by a transversal so that the corresponding angles are congruent, then the lines are parallel. 3-5 Proving Lines Parallel 3-3 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. 50 50 Holt Geometry Alt. Interior Angles Converse: If two lines are cut by a transversal so that the alternate interior angles are congruent, then the lines are parallel. 3-5 Proving Lines Parallel 3-3 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. 60 60 Holt Geometry Alt. Exterior Angles Converse: If two lines are cut by a transversal so that the alternate exterior angles are congruent, then the lines are parallel. 3-5 Proving Lines Parallel 3-3 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. 80 100 Holt Geometry Consecutive Angles Converse: If two lines are cut by a transversal so that the consecutive interior angles are supplementary, then the lines are parallel. 3-5 Proving Lines Parallel 3-3 Example 1: Using the Converse of the Corresponding Angles Postulate A) Use the Converse of the Corresponding Angles Postulate and the given information to show that ℓ || m. 4 8 Yes, ℓ || m because of the Conv. of Corr. s Post. Holt Geometry 3-5 Proving Lines Parallel 3-3 B) 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.) m3 = (4x – 80)°, m7 = (3x – 50)°, x = 30 m3 = 4(30) – 80 = 40 m7 = 3(30) – 50 = 40 3 7 Yes, ℓ || m because of the Conv. of Corr. s Post. Holt Geometry 3-5 Proving Lines Parallel 3-3 C) Use the Converse of the Alt. Int. Angles Postulate and the given information to show that ℓ || m. m7 = (4x + 26)°, m2 = (5x + 12)°, x = 13 m7 = 4(13) + 26 = 78 m2 = 5(13) + 12 = 77 Substitute 13 for x. Substitute 13 for x. m7 = m2 ℓ is not parallel to m because alternate interior angles should be congruent, and these two angles are not. Holt Geometry 3-5 Proving Lines Parallel 3-3 D) Use the Converse of the Alt. Ext. Angles Postulate and the given information to show that ℓ || m. m1 = (3x + 10)°, m8 = (4x - 10)°, x = 20 m1 = 3(20) + 10 = 70 Substitute 13 for x. m8 = 4(20) - 10 = 70Substitute 13 for x. m1 = m8 Yes, ℓ || m because of the converse of alternate exterior angles Holt Geometry 3-5 Proving Lines Parallel 3-3 E) Use the given information and the theorems you have learned to show that r || s. m2 = (10x + 8)°, m3 = (25x – 67)°, x = 5 m2 = 10x + 8 = 10(5) + 8 = 58 m2 + m3 = 58° + 58° = 116° m3 = 25x – 67 = 25(5) – 67 = 58 2 and 3 are same-side interior angles. So, r is not parallel to s because the angles should =180 Holt Geometry 3-5 Proving Lines Parallel 3-3 Example 4: Carpentry Application A carpenter is creating a woodwork pattern and wants two long pieces to be parallel. m1= (8x + 20)° and m2 = (2x + 10)°. If x = 15, show that pieces A and B are parallel. Holt Geometry 3-5 Proving Lines Parallel 3-3 Example 4 Continued m1 = 8x + 20 = 8(15) + 20 = 140 Substitute 15 for x. m2 = 2x + 10 = 2(15) + 10 = 40 m1+m2 = 140 + 40 = 180 Substitute 15 for x. 1 and 2 are supplementary. The same-side interior angles are supplementary, so pieces A and B are parallel by the Converse of the Same-Side Interior Angles Theorem. Holt Geometry 3-5 Proving Lines Parallel 3-3 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. Holt Geometry