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 ∠PRQ

We know that ∠PRQ and ∠SRT are vertical angles. Vertical angles are equal. So ∠PRQ = ∠SRT.

Step2: Calculate the measure of ∠P in △PRQ

The sum of angles in a triangle is \(180^{\circ}\). In \(\triangle PRQ\), if \(\angle Q = 98^{\circ}\) and \(\angle PRQ=\angle SRT\), and in \(\triangle TRS\), \(\angle T = 53^{\circ}\).
For \(\triangle PRQ\), \(\angle P=180^{\circ}-\angle Q - \angle PRQ\). For \(\triangle TRS\), \(\angle S=180^{\circ}-\angle T-\angle SRT\).
Since \(\angle PRQ=\angle SRT\), \(\angle P = 180^{\circ}-98^{\circ}-\angle PRQ\) and \(\angle S=180^{\circ}- 53^{\circ}-\angle SRT\).
But we can also use the AA (angle - angle) similarity criterion.
We know that \(\angle PRQ=\angle SRT\) (vertical angles) and \(\angle Q = 98^{\circ}\), \(\angle T=53^{\circ}\), \(\angle P=180^{\circ}-98^{\circ}-\angle PRQ\), \(\angle S=180^{\circ}-53^{\circ}-\angle SRT\).
Since two angles of \(\triangle PRQ\) and \(\triangle TRS\) are not equal.
Let's calculate the third - angle of \(\triangle PRQ\): \(\angle P=180^{\circ}-98^{\circ}-\angle PRQ\). The third - angle of \(\triangle TRS\): \(\angle S = 180^{\circ}-53^{\circ}-\angle SRT\). Since \(\angle PRQ=\angle SRT\) (vertical angles), \(\angle P=180^{\circ}-98^{\circ}-\angle x\) and \(\angle S=180^{\circ}-53^{\circ}-\angle x\) (where \(\angle x=\angle PRQ=\angle SRT\)). \(\angle P = 82^{\circ}-\angle x\) and \(\angle S=127^{\circ}-\angle x\)

Answer:

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