10 12-11 holt 3-3 proving lines parallel

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

Example 1A: Using the Converse of the Corresponding Angles PostulateUse 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 || mConv. of Corr. s Post.

Example 1B: Using the Converse of the Corresponding Angles PostulateUse 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 = 30m3 = 4(30) – 80 = 40Substitute 30 for x.m7 = 3(30) – 50 = 40 Substitute 30 for x.m3 = m7Trans. Prop. of Equality3  7Def. of  s.ℓ || m Conv. of Corr. s Post.

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4 8 Congruent angles4 8 4 and 8 are alternate exterior angles.r || sConv. of Alt. Ext. s Thm.

Example 3: Proving Lines ParallelGiven:ℓ || m, 1 3Prove: p || r

Example 3 Continued1. Given1. ℓ ||m2.1  22.Corr. s Post.3.1  33. Given4.2  34. Trans. Prop. of 5.p || r5. Conv. of Alt. Int. s Thm.

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

Check It Out! Example 3 Continued 1. Given1.1  42. m1 = m42.Def. s 3.3 and4 are supp.3.Given4. m3 + m4 = 1804.Def. Supp. s5. m3 + m1 = 1805. Substitution6. m2 = m36. Vert.s Thm.7. m2 + m1 = 1807. Substitution8. ℓ || m8. Conv. of Same-Side Interior sPost.

Example 4: Carpentry ApplicationA 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 ContinuedA line through the center of the horizontal piece forms a transversal to pieces A and B.1 and 2 are same-side interior angles. If 1 and 2 are supplementary, then pieces A and B are parallel.Substitute 15 for x in each expression.

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

Lesson Quiz: Part IName the postulate or theoremthat proves p || r.1. 4 5Conv. of Alt. Int. sThm.2. 2 7Conv. of Alt. Ext. sThm.3. 3 7Conv. of Corr. sPost.4. 3 and 5 are supplementary.Conv. of Same-Side Int. sThm.

10 12-11 holt 3-3 proving lines parallel

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10 12-11 holt 3-3 proving lines parallel