# Honors Geometry Companion Book, Volume 1

3.1.2 Angles, Parallel Lines, and Transversals (continued)

The process for finding m ∠ UTQ is very similar to the process used to find m ∠ STQ in Example 2. Again, first classify the pair ∠ UTQ and ∠ RQT . Since the two angles are on the same side of the transversal and inside of the other lines, ∠ UTQ and ∠ RQT are same-side interior angles. Therefore, since lines PR and SU are parallel, ∠ UTQ and ∠ RQT are supplementary by the Same-Side Interior Angles Theorem. Use this fact to write an equation, then simplify and solve. Once x is known, substitute that value into the expression for m ∠ UTQ , 5 x − 14, and simplify to find m ∠ UTQ . In this example the m ∠ ABC is found by using the Same-Side Interior Angles Theorem. Since the three lines are parallel and the angles represented by 10 x and 11 x + 12 are same-side interior angles, these two angles are supplementary. It follows that (10 x ) + (11 x + 12) = 180. Simplify and solve the equation to find the value of x . Then, substitute that value for x into the expression for m ∠ ABC , 11 x + 12, to find m ∠ ABC .

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