Question

1. what is the measure of angle p?

Explanation:

Step1: Identify congruent triangles

PQRS is a parallelogram, so $\triangle PRS \cong \triangle RPQ$. Side $QS=9.3$ in, $PS=22$ in, $\angle PSQ=35^\circ$. We use $\triangle PSQ$ for Law of Sines.

Step2: Apply Law of Sines to find $\angle q_1$

Law of Sines: $\frac{\sin(\angle q_1)}{PS} = \frac{\sin(35^\circ)}{QS}$
$\sin(\angle q_1) = \frac{22 \times \sin(35^\circ)}{9.3}$
$\sin(35^\circ) \approx 0.5736$, so $\sin(\angle q_1) \approx \frac{22 \times 0.5736}{9.3} \approx \frac{12.6192}{9.3} \approx 1.3569$
This is impossible, so use $\triangle PQR$: $QR=22$ in, $PR=QS=9.3$ in, $\angle QRS=35^\circ$ (opposite $\angle QPS$). Correct triangle: $\triangle PQS$, $\angle PQS = \angle PRS$, use Law of Sines on $\triangle PRS$: $\frac{\sin(\angle p_1)}{9.3} = \frac{\sin(35^\circ)}{22}$

Step3: Calculate $\sin(\angle p_1)$

$\sin(\angle p_1) = \frac{9.3 \times \sin(35^\circ)}{22} \approx \frac{9.3 \times 0.5736}{22} \approx \frac{5.3345}{22} \approx 0.2425$
$\angle p_1 \approx \arcsin(0.2425) \approx 14^\circ$

Step4: Find $\angle p_2$ (alternate interior angle)

$\angle p_2 = \angle r_2 = 35^\circ$ (parallelogram alternate angles)

Step5: Sum angles for $\angle P$

$\angle P = \angle p_1 + \angle p_2 = 14^\circ + 35^\circ = 49^\circ$

Answer:

$49^\circ$