Question

1. determine the measures of ∠a and ∠c to the nearest tenth of a degree.

Explanation:

Step1: Find $\angle A$ using sine - function

In right - triangle $ABC$ with right - angle at $B$, $\sin A=\frac{BC}{AC}$. Given $BC = 11$ and $AC = 23$. So, $\sin A=\frac{11}{23}$. Then $A=\sin^{- 1}(\frac{11}{23})$.
$A=\sin^{-1}(\frac{11}{23})\approx28.4^{\circ}$

Step2: Find $\angle C$ using the angle - sum property of a triangle

Since the sum of angles in a triangle is $180^{\circ}$ and $\angle B = 90^{\circ}$, $\angle A+\angle B+\angle C=180^{\circ}$. So, $\angle C=180^{\circ}-\angle A - \angle B$. Substituting $\angle B = 90^{\circ}$ and $\angle A\approx28.4^{\circ}$, we get $\angle C=180^{\circ}-90^{\circ}-28.4^{\circ}=61.6^{\circ}$

Answer:

$\angle A\approx28.4^{\circ}$, $\angle C\approx61.6^{\circ}$