Question

what are the angle measures of triangle abc?
○ m∠a = 30°, m∠b = 60°, 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: Check the Pythagorean theorem

For a triangle with sides \(a = 12\), \(b=12\sqrt{3}\), \(c = 24\)

$$\begin{align*} a^{2}+b^{2}&=12^{2}+(12\sqrt{3})^{2}\\ &=144 + 432\\ &=576 \end{align*}$$

And \(c^{2}=24^{2} = 576\)
So \(a^{2}+b^{2}=c^{2}\), by the Pythagorean theorem, \(\angle A=90^{\circ}\)

Step2: Use the sine function

We know that \(\sin B=\frac{\text{opposite}}{\text{hypotenuse}}\)
For \(\angle B\), the opposite side is \(AC = 12\sqrt{3}\), hypotenuse \(BC = 24\)
\(\sin B=\frac{12\sqrt{3}}{24}=\frac{\sqrt{3}}{2}\), so \(m\angle B = 60^{\circ}\)

Step3: Use the angle - sum property of a triangle

Since the sum of angles in a triangle is \(180^{\circ}\), \(\angle A+\angle B+\angle C=180^{\circ}\)
\(\angle C=180^{\circ}-\angle A - \angle B=180^{\circ}-90^{\circ}-60^{\circ}=30^{\circ}\)

Answer:

\(m\angle A = 90^{\circ},m\angle B = 60^{\circ},m\angle C = 30^{\circ}\) (the second option)