Question

triangle stu is formed by connecting the midpoints of the side of triangle pqr. the measures of the interior angles of triangle pqr are shown. find the measure of ∠qst. figures not necessarily drawn to scale.

Explanation:

Step1: Find ∠PRQ (wait, no, first find ∠PQR? Wait, triangle PQR has angles at R: 54°, at Q: 82°, at P: 44°? Wait, sum of angles in triangle is 180°. Wait, S is midpoint? Wait, STU is formed by midpoints, so ST is midline? Wait, no, first, let's recall that in triangle PQR, angles are ∠R = 54°, ∠Q = 82°, ∠P = 44° (since 54 + 82 + 44 = 180). Now, S is midpoint of PQ? Wait, no, the problem says STU is formed by connecting midpoints of PQR. So S, T, U are midpoints. So ST is parallel to PR? Wait, no, midline theorem: the segment connecting midpoints of two sides is parallel to the third side and half its length. So if S is midpoint of PQ, T is midpoint of QR, then ST || PR and ST = ½ PR. Then ∠QST would be equal to ∠QPR (corresponding angles) because ST || PR. Wait, ∠QPR is ∠P, which is 44°? Wait, no, wait: ∠QST and ∠QPR: if ST || PR, then ∠QST = ∠QPR (corresponding angles). Wait, but let's check the angles. Wait, triangle PQR: angles at R: 54°, Q: 82°, so angle at P is 180 - 54 - 82 = 44°. Now, S is midpoint of PQ? Wait, no, S is on PQ? Wait, the diagram: S is on PQ, T is on QR, U is on PR? Wait, maybe ST is parallel to PR. So ∠QST = ∠QPR (since ST || PR, corresponding angles). So ∠QST = 44°? Wait, no, wait: angle at P is 44°, so if ST is parallel to PR, then ∠QST = ∠QPR = 44°? Wait, no, maybe I got the sides wrong. Wait, let's re-examine. The triangle PQR: vertices P, Q, R. S is on PQ, T is on QR, U is on PR. Then ST connects midpoints of PQ and QR? No, midpoints: so S is midpoint of PQ, T is midpoint of QR, U is midpoint of PR. Then ST is midline, so ST || PR, and ∠QST = ∠QPR (corresponding angles). So ∠QPR is 44°, so ∠QST = 44°? Wait, but let's confirm. Alternatively, maybe S is midpoint of PQ, so QS = SP. But the key is that ST is midline, so parallel to PR, so ∠QST = ∠QPR (since ST || PR, transversal QP). So ∠QPR is 44°, so ∠QST is 44°? Wait, but let's check the angle sum. 54 (∠R) + 82 (∠Q) + 44 (∠P) = 180, correct. So ∠QST = ∠P = 44°? Wait, no, wait: if ST is parallel to PR, then ∠QST and ∠QPR are corresponding angles, so they should be equal. So ∠QST = 44°. Wait, but maybe I made a mistake. Wait, another approach: since S and T are midpoints, ST is midline, so ST || PR, so ∠QST = ∠QPR (corresponding angles). ∠QPR is 44°, so ∠QST is 44°.

Step2: Confirm angle sum. In triangle PQR, angles are 54 (R), 82 (Q), 44 (P). Sum is 54+82=136, 180-136=44. Correct. Then ST || PR, so corresponding angles ∠QST and ∠QPR are equal. So ∠QST = 44°.

Answer:

44°