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: Check for right triangle

First, we check if triangle \(ABC\) is a right triangle by verifying the Pythagorean theorem. The sides are \(AB = 6\) in, \(AC = 6\sqrt{3}\) in, and \(BC = 12\) in. Let's check \(AB^{2}+AC^{2}\):
\(AB^{2}=6^{2} = 36\)
\(AC^{2}=(6\sqrt{3})^{2}=36\times3 = 108\)
\(AB^{2}+AC^{2}=36 + 108=144\)
And \(BC^{2}=12^{2}=144\). So, \(AB^{2}+AC^{2}=BC^{2}\), which means \(\angle A = 90^{\circ}\) (by Pythagorean theorem converse).

Step2: Identify side - angle relationships

In a right triangle, the side opposite the \(30^{\circ}\) angle is half the hypotenuse. The hypotenuse \(BC = 12\) in, and \(AB = 6\) in, which is half of \(12\) in. So, the angle opposite \(AB\) is \(\angle C\). So, \(m\angle C=30^{\circ}\).

Step3: Find the remaining angle

Since the sum of angles in a triangle is \(180^{\circ}\), and we know \(\angle A = 90^{\circ}\) and \(\angle C = 30^{\circ}\), then \(m\angle B=180^{\circ}-90^{\circ}-30^{\circ}=60^{\circ}\).

Answer:

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