Question

problem 11: (first taught in lesson 31) complete this sentence by selecting always, sometimes, or never. the measure of an exterior angle of a triangle __________ equals the sum of the measures of the two remote interior angles. after you pick your answer press go. a. always b. sometimes c. never

Explanation:

Step1: Recall angle - sum property

The sum of the interior angles of a triangle is 180 degrees. Let the triangle have interior angles \(A\), \(B\), and \(C\), so \(A + B + C=180^{\circ}\). Let an exterior angle be \(E\) adjacent to angle \(C\). Then \(E + C = 180^{\circ}\) (linear - pair of angles).

Step2: Solve for the exterior angle

From \(E + C=180^{\circ}\) and \(A + B + C = 180^{\circ}\), we can substitute \(180^{\circ}-E\) for \(C\) in the second equation: \(A + B+(180^{\circ}-E)=180^{\circ}\). Simplifying gives \(E=A + B\), where \(A\) and \(B\) are the two remote interior angles to the exterior angle \(E\). So the measure of an exterior angle of a triangle always equals the sum of the measures of the two remote interior angles.

Answer:

A. always