Question

solving for all unknown measures in a triangle. determine the measures of all unknown angles and side lengths of △pqr. round side lengths to the nearest hundredth. m∠r =
pr =
pq =

Explanation:

Step1: Find angle R

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

Step2: Use the Law of Sines to find PR

By the Law of Sines, $\frac{PR}{\sin Q}=\frac{QR}{\sin P}$. We know $QR = 11$, $\angle Q=31^{\circ}$, $\angle P = 99^{\circ}$. So, $PR=\frac{11\times\sin31^{\circ}}{\sin99^{\circ}}$.
$PR=\frac{11\times0.5150}{0.9877}\approx5.73$

Step3: Use the Law of Sines to find PQ

By the Law of Sines, $\frac{PQ}{\sin R}=\frac{QR}{\sin P}$. We know $QR = 11$, $\angle R = 50^{\circ}$, $\angle P=99^{\circ}$. So, $PQ=\frac{11\times\sin50^{\circ}}{\sin99^{\circ}}$.
$PQ=\frac{11\times0.7660}{0.9877}\approx8.48$

Answer:

$m\angle R = 50^{\circ}$, $PR\approx5.73$, $PQ\approx8.48$