Question

given circle ( p ) with radii ( overline{pr}, overline{pq} ), and ( overline{ps} ) and chords ( overline{tr} ) and ( overline{qs}, overline{tr} parallel overline{qs} ). name an angle that has the same measure as ( angle prt ).

Explanation:

Step1: Use the property of parallel lines

Since \(\overline{TR}\parallel\overline{QS}\), we can use the alternate - interior angles theorem. When two parallel lines are cut by a transversal, alternate - interior angles are congruent. Here, the transversal is \(PQ\) (or \(PR\) related to the parallel lines \(\overline{TR}\) and \(\overline{QS}\)).
Let's consider the transversal \(PQ\) (or the relationship between the lines and the circle). In the circle, \(PR = PQ=PS\) (radii of the circle).
The alternate - interior angles for the parallel lines \(\overline{TR}\) and \(\overline{QS}\) with respect to the transversal (we can think of the line that relates the angles).
Since \(\overline{TR}\parallel\overline{QS}\), \(\angle PRT\) and \(\angle PQS\) are alternate - interior angles.

Answer:

\(\angle PQS\)