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 angle at P in ΔPQR

In triangle \( PQR \), the sum of interior angles is \( 180^\circ \). Given \( \angle Q = 82^\circ \), \( \angle R = 54^\circ \), so \( \angle P = 180^\circ - 82^\circ - 54^\circ = 44^\circ \) (matches the given \( \angle P = 44^\circ \)).

Step2: Analyze midsegments (ST, TU, US)

Since \( S, T, U \) are midpoints, \( ST \parallel PR \), \( TU \parallel PQ \), \( US \parallel QR \) (by Midline Theorem: midsegment is parallel to third side).

Step3: Find \( \angle QST \)

\( ST \parallel PR \), so \( \angle QST = \angle P \) (corresponding angles, as \( ST \parallel PR \) and \( PQ \) is transversal). Thus, \( \angle QST = 44^\circ \)? Wait, no—wait, \( S \) is midpoint of \( PQ \), \( T \) midpoint of \( QR \). Wait, maybe better: \( \triangle QST \) and \( \triangle QPR \) are similar (by SAS similarity, since \( QS/ QP = QT/ QR = 1/2 \), and included angle \( \angle Q \) is common). So corresponding angles: \( \angle QST = \angle QPR = 44^\circ \)? Wait, no, wait the diagram: \( \angle P = 44^\circ \), and \( ST \parallel PR \), so \( \angle QST = \angle P = 44^\circ \)? Wait, no, let's recheck.

Wait, in \( \triangle PQR \), angles: \( \angle Q = 82^\circ \), \( \angle R = 54^\circ \), \( \angle P = 44^\circ \). \( S \) is midpoint of \( PQ \), \( T \) midpoint of \( QR \), so \( ST \) is midline, so \( ST \parallel PR \), so \( \angle QST = \angle QPR = 44^\circ \)? Wait, no, corresponding angles: when \( ST \parallel PR \), transversal \( PQ \), so \( \angle QST \) and \( \angle QPR \) are corresponding? Wait, \( S \) is on \( PQ \), \( T \) on \( QR \), so \( ST \) connects midpoints, so \( ST \parallel PR \), so \( \angle QST = \angle P \) (since \( PQ \) is transversal, \( \angle QST \) and \( \angle P \) are corresponding angles). Thus, \( \angle QST = 44^\circ \)? Wait, but let's check another way. In \( \triangle QST \), \( \angle Q = 82^\circ \), if we can find another angle. Wait, maybe I made a mistake. Wait, \( TU \parallel PQ \), so \( \angle TUS = \angle Q = 82^\circ \)? No, maybe better: since \( ST \parallel PR \), \( \angle QST = \angle P = 44^\circ \). Wait, the given \( \angle P = 44^\circ \), so yes, corresponding angles. So \( \angle QST = 44^\circ \)? Wait, but let's recalculate \( \angle P \): \( 180 - 82 - 54 = 44 \), correct. So \( ST \parallel PR \), so \( \angle QST = \angle P = 44^\circ \).

Wait, but maybe the answer is \( 44^\circ \)? Wait, no, wait the diagram: \( \angle R = 54^\circ \), \( \angle Q = 82^\circ \), \( \angle P = 44^\circ \). \( S \) is midpoint of \( PQ \), \( T \) midpoint of \( QR \), so \( ST \parallel PR \), so \( \angle QST = \angle P = 44^\circ \). So the measure of \( \angle QST \) is \( 44^\circ \).

Answer:

\( 44^\circ \)