Question

given circle q with radii \\( \overline{q s}, \overline{q r} \\), and \\( \overline{q t} \\) and chords \\( \overline{u s} \\) and \\( \overline{r t}, \overline{u s} \parallel \overline{r t} \\). name an angle that has the same measure as \\( \angle q t r \\).

Explanation:

Step1: Use properties of parallel lines and radii

Since \( \overline{US}\parallel\overline{RT}\), and \(QS = QT\) (radii of the same circle), consider the alternate - interior angles.
When two parallel lines are cut by a transversal, alternate - interior angles are equal. Here, \( \overline{QS}\) is a transversal for \( \overline{US}\) and \( \overline{RT}\).
We know that \( \angle QUS=\angle QTR\) (alternate - interior angles for \( \overline{US}\parallel\overline{RT}\) and transversal \( \overline{QT}\)). Also, \( \triangle QUS\) and \( \triangle QTR\): \(QS = QT\) (radii), \(QU = QR\) (radii). But a simpler approach is using the property of parallel lines and the fact that \( \angle QSU=\angle QTR\) (alternate - interior angles for \( \overline{US}\parallel\overline{RT}\) and transversal \( \overline{ST}\)).

Answer:

\(\angle QSU\)