Question

in the picture below, line pq is parallel to line rs, and the lines are cut by a transversal, line tu. the transversal is not perpendicular to the parallel lines.
note: figure is not drawn to scale.
which of the following are supplementary exterior angles?
a. ∠e and ∠f
b. ∠h and ∠g
c. ∠h and ∠e
d. ∠x and ∠e

Explanation:

Step1: Recall the definition of supplementary angles

Supplementary angles are two angles whose sum is \(180^{\circ}\).

Step2: Analyze each option

• Option A: \(\angle E\) and \(\angle F\) are adjacent angles forming a linear - pair. By the linear - pair postulate, \(\angle E+\angle F = 180^{\circ}\). But they are interior angles (not exterior).
• Option B: \(\angle H\) and \(\angle G\) are adjacent angles forming a linear - pair. \(\angle H+\angle G=180^{\circ}\), but they are interior angles (not exterior).
• Option C: \(\angle H\) and \(\angle E\) are not adjacent. Since \(PQ\parallel RS\) and \(TU\) is a transversal, \(\angle H\) and \(\angle E\) are not in a position to be supplementary.
• Option D: \(\angle X\) and \(\angle E\):
• \(\angle X\) and \(\angle G\) are vertical angles (\(\angle X=\angle G\)).
• Since \(PQ\parallel RS\), \(\angle G\) and \(\angle E\) are same - side exterior angles.
• By the same - side exterior angles theorem, for parallel lines \(PQ\) and \(RS\) cut by transversal \(TU\), \(\angle G+\angle E = 180^{\circ}\). Since \(\angle X=\angle G\), then \(\angle X+\angle E=180^{\circ}\)

Answer:

D. \(\angle X\) and \(\angle E\)