Question

△prq and △trs are shown below. which statement is true? △prq is similar to △trs. △prq is not similar to △trs. there is not enough information to determine whether the triangles are similar.

Explanation:

Step1: Find the measure of \(\angle PRQ\) and \(\angle SRT\)

Since \(\angle PRQ\) and \(\angle SRT\) are vertical angles, \(\angle PRQ=\angle SRT\) (vertical angles are equal).

Step2: Calculate the third angle of \(\triangle PRQ\)

In \(\triangle PRQ\), using the angle - sum property of a triangle (\(\angle P+\angle Q+\angle PRQ = 180^{\circ}\)). Let \(\angle PRQ=\angle SRT = x\). We know \(\angle Q = 98^{\circ}\). So \(\angle P=180^{\circ}-\angle Q-\angle PRQ=180^{\circ}-98^{\circ}-x = 82^{\circ}-x\).
In \(\triangle TRS\), using the angle - sum property (\(\angle S+\angle T+\angle SRT=180^{\circ}\)), \(\angle T = 53^{\circ}\), \(\angle S=180^{\circ}-\angle T-\angle SRT=180^{\circ}-53^{\circ}-x=127^{\circ}-x\).
Since \(\angle Q
eq\angle T\) and \(\angle P
eq\angle S\) (because \(98
eq53\) and \(82 - x
eq127 - x\)) and we only have one pair of equal angles (\(\angle PRQ=\angle SRT\)), the two triangles do not satisfy the AA (angle - angle) similarity criterion (which requires two pairs of equal angles for triangle similarity).

Answer:

\(\triangle PRQ\) is not similar to \(\triangle TRS\).