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

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Accepted Answer

# Explanation:
## Step1: Find base angles of isosceles triangle SQT
In isosceles triangle $ SQT $, $ \angle STQ = 48^\circ $. Let $ \angle QST=\angle SQT = x $ (since sides $ ST = QT $ from markings, so base angles are equal). Using triangle angle sum: $ 48^\circ + x + x = 180^\circ $.
Simplify: $ 2x = 180^\circ - 48^\circ = 132^\circ $, so $ x = 66^\circ $. Wait, no—wait, maybe $ ST = SQ $? Wait, the diagram has markings: $ SQ $ and $ RQ $ are marked equal, $ ST $ and $ QT $ are marked equal? Wait, let's re-examine. Triangle $ SQT $ is isosceles with $ ST = QT $ (markings on $ ST $ and $ QT $), so $ \angle QST=\angle SQT $. Wait, no, $ \angle STQ = 48^\circ $, so the two equal angles are $ \angle QST $ and $ \angle SQT $. Wait, no, angle at $ T $ is $ 48^\circ $, so the other two angles: $ (180 - 48)/2 = 66^\circ $. But then, $ TR $ is a segment from $ T $ to $ R $ on $ SQ $, with $ SR = RQ $ (markings on $ SQ $), so $ R $ is the midpoint, and $ TR $ is a median? Wait, no—maybe $ \triangle STR $ and $ \triangle QTR $ are congruent? Wait, maybe I misread. Wait, the problem is to find $ \angle STR $. Let's correct: In $ \triangle SQT $, $ ST = QT $ (isosceles), so $ \angle S = \angle Q $. Then, $ \angle STQ = 48^\circ $, so $ \angle S + \angle Q = 180 - 48 = 132^\circ $, so $ \angle S = \angle Q = 66^\circ $. Now, $ R $ is the midpoint of $ SQ $ (since $ SR = RQ $), so $ TR $ is the median, and in an isosceles triangle, the median from $ T $ to $ SQ $ is also the angle bisector. So $ \angle STR = \frac{1}{2} \angle STQ $? Wait, no—wait, $ \angle STQ = 48^\circ $, and $ TR $ bisects $ \angle STQ $? Wait, no, maybe $ \triangle SQT $ has $ SQ = ST $? Wait, the markings: $ SQ $ and $ ST $ have one mark, $ QT $ has one mark? Wait, maybe $ SQ = ST $, so $ \angle STQ = \angle SQT = 48^\circ $. Then, $ \angle QST = 180 - 48 - 48 = 84^\circ $. No, this is confusing. Wait, the answer choices are 24, 38, 48, 76. Let's think again. Wait, maybe $ \triangle SQT $ is isosceles with $ SQ = QT $, so $ \angle QST = \angle QTS = 48^\circ $, then $ \angle SQT = 180 - 48 - 48 = 84^\circ $. No, not matching. Wait, maybe the correct approach: In isosceles $ \triangle SQT $, $ ST = SQ $, so $ \angle STQ = \angle SQT = 48^\circ $, then $ \angle QST = 180 - 48 - 48 = 84^\circ $. Then $ TR $ is a median, so $ SR = RQ $, and $ ST = SQ $, so $ \triangle STR $ is isosceles? No. Wait, maybe the diagram shows that $ TR $ is perpendicular? No. Wait, the key is that $ \angle STR $ is half of $ \angle STQ $? Wait, 48/2=24? But 24 is an option. Wait, maybe $ \triangle SQT $ has $ ST = QT $, so $ \angle S = \angle Q $, $ \angle STQ = 48^\circ $, so $ \angle S = \angle Q = (180 - 48)/2 = 66^\circ $. Then, $ TR $ is the angle bisector? No, maybe $ R $ is the midpoint, and $ TR $ is the median, so in $ \triangle SQT $, median $ TR $ splits $ \angle STQ $ into two equal angles? Wait, no, in an isosceles triangle, the median from the apex (angle at $ T $) is the angle bisector. So $ \angle STR = \angle QTR = 48^\circ / 2 = 24^\circ $? Wait, but 24 is an option. Wait, maybe that's it. So step by step:

1. Triangle $ SQT $ is isosceles with $ ST = QT $ (markings on $ ST $ and $ QT $), so $ \angle QST = \angle SQT $.
2. Angle sum: $ \angle STQ + \angle QST + \angle SQT = 180^\circ $.
3. Given $ \angle STQ = 48^\circ $, so $ 48^\circ + 2\angle QST = 180^\circ $ → $ 2\angle QST = 132^\circ $ → $ \angle QST = 66^\circ $. Wait, no, this contradicts. Wait, maybe $ SQ = ST $, so $ \angle STQ = \angle SQT = 48^\circ $, then $ \angle QST = 180 - 48 - 48 = 84^\circ $. Then $ TR $ is the median, so $ SR = RQ $, and $ ST = SQ $, so $ \triangle STR $ has $ ST = SQ = 2SR $, so $ ST = 2SR $, making $ \angle STR = 30^\circ $? No, not matching. Wait, the answer choices include 38? Wait, maybe I made a mistake. Wait, let's check the answer options. The options are 24, 38, 48, 76. Let's recalculate: If $ \triangle SQT $ is isosceles with $ SQ = QT $, then $ \angle QST = \angle QTS = 48^\circ $, so $ \angle SQT = 180 - 48 - 48 = 84^\circ $. Then $ TR $ bisects $ \angle SQT $? No. Wait, maybe the diagram has $ ST = SQ $ and $ SR = RQ $, so $ TR $ is the median, and in $ \triangle SQT $, $ ST = SQ $, so it's isosceles with $ ST = SQ $, angle at $ S $ is $ \angle QST $, angle at $ T $ is $ \angle STQ = 48^\circ $, angle at $ Q $ is $ \angle SQT $. Wait, no, sides: $ ST $ and $ SQ $ are equal (one mark), $ QT $ has one mark? Wait, maybe $ ST = SQ = QT $, making it equilateral? No, angle is 48. Wait, I think the correct approach is: In isosceles $ \triangle SQT $, $ \angle STQ = 48^\circ $, and $ TR $ is the angle bisector (since $ R $ is the midpoint of $ SQ $, and in isosceles triangle, median is angle bisector). So $ \angle STR = \frac{1}{2} \angle STQ $? No, $ 48/2 = 24 $, which is option A. So that must be it.

## Step2: Calculate $ \angle STR $
Since $ TR $ is the angle bisector (median in isosceles triangle), $ \angle STR = \frac{1}{2} \angle STQ $.
$ \angle STQ = 48^\circ $, so $ \angle STR = \frac{48^\circ}{2} = 24^\circ $.

# Answer:
$ 24^\circ $ (Option: 24°)

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Step1: Find base angles of isosceles triangle SQT

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