Question

what are the angle measures in triangle abc?
options:

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

triangle abc has sides: bc = 12 in, ab = 6 in, ac = ( 6sqrt{3} ) in.

Explanation:

Step1: Verify right triangle

Check via Pythagorean theorem:
$$6^2 + (6\sqrt{3})^2 = 36 + 108 = 144 = 12^2$$

Step2: Match sides to angles

Longest side (12 in) = hypotenuse, opposite $\angle A$. So $m\angle A=90^\circ$. Shortest side (6 in) opposite $\angle C$, so $m\angle C=30^\circ$. Remaining angle: $180^\circ-90^\circ-30^\circ=60^\circ$, so $m\angle B=60^\circ$.

Answer:

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