3.1.3 Proving that Lines are Parallel (continued)

In this example, expressions for the measures of angles 2, 4, and 6 are given. Use these expressions to prove that p || q . Notice that ∠ 2 and ∠ 4 are vertical angles. Therefore, the m ∠ 2 = m ∠ 4. Use this fact to write an equation and solve for z . Notice that ∠ 4 and ∠ 6 are alternate interior angles. So, if ∠ 4 and ∠ 6 are congruent, then it can be concluded that p || q by the Converse of the Alternate Interior Angles Theorem. Substitute the value of z into the expressions for m ∠ 4 and m ∠ 6. Since m ∠ 4 = 70 ° and m ∠ 6 = 70 ° , ∠ 4 ≅ ∠ 6. Therefore, since these alternate interior angles are congruent, p || q by the Converse of the Alternate Interior Angles Theorem.

Example 2 Using the Converse of the Same Side Interior Angles Theorem By the converse of the Same-Side Interior Angles

Theorem, if two coplanar lines are cut by a transversal so that a pair of same-side interior angles are supplementary, then the two lines are parallel.

In this example, two lines, m and n , are proven to be parallel by using the Converse of the Same-Side Interior Angles Theorem. Expressions for the measures of angles 3 and 8 are given. Notice that ∠ 3 and ∠ 8 are same-side interior angles. If it can be shown that ∠ 3 and ∠ 8 are supplementary, then it can be concluded that m || n by the Converse of the Same-Side Interior Angles Theorem. Remember, the sum of two supplementary angles is 180 ° .

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