According to the $\mathrm{Law} \mathrm{of} \mathrm{Sines}$, the measure of an exterior angle of a triangle is equal to which of the following measures?

• A. The sum of the measures of the two non-adjacent interior angles $(\alpha +\beta )$
• B. The difference between the measures of the two adjacent interior angles $|\alpha -\beta |$
• C. Half the measure of the adjacent interior angle $\frac{1}{2}\alpha$
• D. The product of the measures of the two non-adjacent interior angles $\alpha \beta$

Answer: (A)

Verified step by step guidance

1. Recall the definition of an exterior angle of a triangle: it is formed by extending one side of the triangle, creating an angle adjacent to an interior angle.
2. Understand the Exterior Angle Theorem, which states that the measure of an exterior angle of a triangle is equal to the sum of the measures of the two non-adjacent (or remote) interior angles.
3. Identify the two non-adjacent interior angles relative to the exterior angle in question. These are the interior angles that are not adjacent to the exterior angle.
4. Express this relationship mathematically as: \(\text{Exterior Angle} = \text{Interior Angle}_1 + \text{Interior Angle}_2\), where \(\text{Interior Angle}_1\) and \(\text{Interior Angle}_2\) are the two non-adjacent interior angles.
5. Conclude that among the given options, the correct one is the sum of the measures of the two non-adjacent interior angles, as per the Exterior Angle Theorem.