Question

what are the angle measures in triangle abc? m∠a = 60°, m∠b = 30°, m∠c = 90°. m∠a = 90°, m∠b = 60°, m∠c = 30°. m∠a = 60°, m∠b = 90°, m∠c = 30°. m∠a = 90°, m∠b = 30°, m∠c = 60°.

Explanation:

Step1: Use cosine - law

$\cos A=\frac{AB^{2}+AC^{2}-BC^{2}}{2\cdot AB\cdot AC}$. Here, $AB = 6$, $AC=6\sqrt{3}$, $BC = 12$.
$AB^{2}=36$, $AC^{2}=108$, $BC^{2}=144$.
$\cos A=\frac{36 + 108-144}{2\times6\times6\sqrt{3}}=0$, so $m\angle A = 90^{\circ}$.

Step2: Use sine - law

$\frac{BC}{\sin A}=\frac{AC}{\sin B}$. Since $\sin A = 1$, $\sin B=\frac{AC}{BC}=\frac{6\sqrt{3}}{12}=\frac{\sqrt{3}}{2}$, so $m\angle B = 60^{\circ}$.

Step3: Find $\angle C$

$m\angle C=180^{\circ}-m\angle A - m\angle B=30^{\circ}$.

Answer:

$m\angle A = 90^{\circ},m\angle B = 60^{\circ},m\angle C = 30^{\circ}$