Question

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

Explanation:

Step1: Recall right - triangle ratio

In a 30 - 60 - 90 triangle, the side lengths are in the ratio $1:\sqrt{3}:2$. If the shortest side (opposite the 30 - degree angle) is $a$, the side opposite the 60 - degree angle is $a\sqrt{3}$, and the hypotenuse is $2a$.

Step2: Identify side lengths

In triangle $ABC$, we have side lengths $12$, $12\sqrt{3}$, and $24$. Here, $a = 12$, $a\sqrt{3}=12\sqrt{3}$, and $2a = 24$.

Step3: Determine angle measures

The side of length $12$ is opposite the smallest angle. The angle opposite the side of length $12$ is $30^{\circ}$, the angle opposite the side of length $12\sqrt{3}$ is $60^{\circ}$, and the hypotenuse is $24$. The right - angle (90°) is opposite the hypotenuse. Since the side of length $12$ is opposite $\angle C$, $\angle C=30^{\circ}$, the side of length $12\sqrt{3}$ is opposite $\angle A$, $\angle A = 60^{\circ}$, and the side of length $24$ is opposite $\angle B$, $\angle B=90^{\circ}$.

Answer:

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