3.1.2 Angles, Parallel Lines, and Transversals (continued)

In this example, the measure of ∠ UTW is given to be 74 ° and the measure of ∠ PQT is to be found. Notice that ∠ UTW and ∠ PQT are alternate interior angles. Furthermore, notice that line PR is given to be parallel to line SU . Since line PR || line SU , the Alternate Interior Angles Theorem can be applied. The Alternate Interior Angles Theorem states that if two parallel lines are cut by a transversal, then the alternate interior angles are congruent. Therefore, by the Alternate Interior Angles Theorem, m ∠ UTW = m ∠ PQT . So, since m ∠ UTW = 74 ° , m ∠ PQT = 74 ° as well. Notice that the lines PR and SU are given to be parallel, that expressions for the measures of ∠ STQ and ∠ PQW are given, and that m ∠ STQ is to be found. First, classify the pair ∠ STQ and ∠ PQW . Since ∠ STQ and ∠ PQW are in the same relative position in each group of 4 angles, ∠ STQ and ∠ PQW are corresponding angles. So, since the lines are parallel and ∠ STQ and ∠ PQW are corresponding angles, the Corresponding Angles Postulate can be applied, resulting in the fact that m ∠ STQ = m ∠ PQW . Since m ∠ STQ = m ∠ PQW , it follows that 2 a + 30 = 3 a − 8. Solve this equation to find the value of a . Once a is known, substitute that value into the expression for m ∠ STQ , 2 a + 30, and simplify.

Example 2 Using the Same-Side Interior Angles Theorem

If two parallel lines are cut by a transversal, then the two pairs of same-side interior angles are supplementary. Same-side interior angles are on the same side of the transversal and on the inside of the other two lines. There are two pairs of same-side interior angles.

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