Question

determining angle/side relationships
what are the angle measures in triangle abc?

• ( mangle a = 90^circ, mangle b = 30^circ, mangle c = 60^circ )
• ( mangle a = 60^circ, mangle b = 90^circ, mangle c = 30^circ )
• ( mangle a = 90^circ, mangle b = 60^circ, mangle c = 30^circ )
• ( mangle a = 60^circ, mangle b = 30^circ, mangle c = 90^circ )

Explanation:

Step1: Verify right triangle

Check if sides satisfy Pythagoras:
$$AB^2 + AC^2 = 6^2 + (6\sqrt{3})^2 = 36 + 108 = 144$$
$$BC^2 = 12^2 = 144$$
So $\triangle ABC$ is right-angled at $A$, $m\angle A=90^\circ$.

Step2: Find $\angle B$ via sine

Use $\sin(\angle B)=\frac{AC}{BC}$:
$$\sin(\angle B)=\frac{6\sqrt{3}}{12}=\frac{\sqrt{3}}{2}$$
Thus $m\angle B=60^\circ$.

Step3: Calculate $\angle C$

Sum of angles in triangle is $180^\circ$:
$$m\angle C=180^\circ - 90^\circ - 60^\circ=30^\circ$$

Answer:

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