Question

solving for all unknown measures in a triangle
determine the measures of all unknown angles and side lengths of $delta pqr$. round side lengths to the nearest hundredth.
$mangle r = square^circ$
$pr approx square$
$pq approx square$
in $delta pqr$, $angle p = 99^circ$, $angle q = 31^circ$, side $qr = 11$

Explanation:

Step1: Calculate $\angle R$

The sum of angles in a triangle is $180^\circ$.
$$m\angle R = 180^\circ - 99^\circ - 31^\circ = 50^\circ$$

Step2: Find length $PR$

Use the Law of Sines: $\frac{PR}{\sin Q} = \frac{QR}{\sin P}$
$$PR = \frac{QR \cdot \sin Q}{\sin P} = \frac{11 \cdot \sin 31^\circ}{\sin 99^\circ}$$
$$\sin 31^\circ \approx 0.5150, \sin 99^\circ \approx 0.9877$$
$$PR \approx \frac{11 \cdot 0.5150}{0.9877} \approx 5.73$$

Step3: Find length $PQ$

Use the Law of Sines: $\frac{PQ}{\sin R} = \frac{QR}{\sin P}$
$$PQ = \frac{QR \cdot \sin R}{\sin P} = \frac{11 \cdot \sin 50^\circ}{\sin 99^\circ}$$
$$\sin 50^\circ \approx 0.7660$$
$$PQ \approx \frac{11 \cdot 0.7660}{0.9877} \approx 8.53$$

Answer:

$m\angle R = 50^\circ$
$PR \approx 5.73$
$PQ \approx 8.53$