these figures are congruent. what is ( mangle q )?

Explanation:

Step1: Identify congruent side pairs

Match equal sides: $AC = QR = 67$ in, $PR = 74$ in, $PQ = 50$ in. So $\triangle ABC \cong \triangle PQR$ with correspondence $A\leftrightarrow P$, $C\leftrightarrow R$, $B\leftrightarrow Q$.

Step2: Find $\angle B$ first

Sum of angles in triangle is $180^\circ$. For $\triangle ABC$: $\angle B = 180^\circ - \angle A - \angle C$. First calculate $\angle A$ from $\triangle PQR$: $\angle A = \angle P = 62^\circ$, $\angle C = 77^\circ$. $\angle B = 180^\circ - 62^\circ - 77^\circ = 41^\circ$.

Step3: Match congruent angles

Since $\angle Q = \angle B$ (corresponding angles of congruent triangles).

Answer:

$41^\circ$