Question

triangle sqt is isosceles. the measure of angle stq is 48°. what is the measure of ∠str? 24° 38° 48° 76°

Explanation:

Step1: Analyze isosceles triangle SQT

In isosceles triangle \( SQT \), \( ST = QT \) (marked with tick marks), so \( \angle S=\angle Q \). The sum of angles in a triangle is \( 180^\circ \). Given \( \angle STQ = 48^\circ \), we find \( \angle S+\angle Q=180 - 48=132^\circ \). Since \( \angle S=\angle Q \), each is \( \frac{132}{2} = 66^\circ \). Wait, no, maybe \( SR = RQ \) and \( TR \) is a median? Wait, the diagram shows \( SR = RQ \) (tick marks on \( SQ \)) and \( ST = TQ \)? Wait, no, the tick marks on \( ST \) and \( TQ \)? Wait, the problem says triangle \( SQT \) is isosceles. Let's re - examine. If \( ST = TQ \), then \( \angle S=\angle Q \). But maybe \( TR \) bisects \( \angle STQ \)? Wait, the angle \( \angle STQ = 48^\circ \), and if \( TR \) is the angle bisector (since \( SR = RQ \), maybe \( TR \) is the median and angle bisector in isosceles triangle). So \( \angle STR=\frac{1}{2}\angle STQ \)? Wait, no, that would be \( 24^\circ \), but wait, maybe I made a mistake. Wait, let's calculate the base angles. In triangle \( SQT \), \( \angle STQ = 48^\circ \), so the other two angles: \( \angle S+\angle Q=180 - 48 = 132^\circ \). If \( ST = TQ \), then \( \angle S=\angle Q = 66^\circ \). But then, if \( R \) is the mid - point of \( SQ \), then \( TR \) is the altitude, median, and angle bisector. So \( \angle STR=\frac{1}{2}\angle STQ \)? Wait, no, \( \angle STQ = 48^\circ \), so if \( TR \) bisects it, \( \angle STR = 24^\circ \). But wait, maybe the triangle is isosceles with \( SQ = ST \)? No, the tick marks: looking at the diagram, \( ST \) and \( TQ \) have one tick, \( SR \) and \( RQ \) have one tick. So \( ST=TQ \), \( SR = RQ \). So triangle \( SQT \) is isosceles with \( ST = TQ \), and \( R \) is the mid - point of \( SQ \). Then \( TR \) is the angle bisector of \( \angle STQ \). So \( \angle STR=\frac{1}{2}\times48^\circ = 24^\circ \).

Step2: Confirm the angle measure

Since \( TR \) bisects \( \angle STQ \) (because \( R \) is the mid - point of \( SQ \) in isosceles triangle \( SQT \) with \( ST = TQ \)), we calculate \( \angle STR=\frac{48^\circ}{2}=24^\circ \).

Answer:

\( 24^\circ \) (corresponding to the option with \( 24^\circ \))