Question

determine whether each statement about angle relationships of a triangle is true. select true or false for each statement. a right triangle always has obtuse exterior angles at two vertices. the sum of the measures of exterior angles minus the sum of the measures of interior angles is 180°. the sum of an exterior angle and the two nonadjacent interior angles is 180°. an obtuse triangle always has only one vertex with an acute exterior angle.

Explanation:

Step1: Recall exterior - angle properties

An exterior angle of a triangle is supplementary to its adjacent interior angle. In a right - triangle, one angle is 90°. The other two interior angles are acute. The exterior angles adjacent to the acute interior angles are obtuse. So, a right triangle always has obtuse exterior angles at two vertices. This statement is True.

Step2: Recall sum of interior and exterior angles

The sum of the interior angles of a triangle is 180°. The sum of the exterior angles of any polygon (including a triangle) is 360°. So, \(360 - 180=180\). The sum of the measures of exterior angles minus the sum of the measures of interior angles is 180°. This statement is True.

Step3: Recall exterior - angle theorem

The exterior - angle theorem states that the measure of an exterior angle of a triangle is equal to the sum of the measures of the two non - adjacent interior angles. It is not the case that the sum of an exterior angle and the two nonadjacent interior angles is 180°. This statement is False.

Step4: Analyze obtuse - triangle exterior angles

In an obtuse triangle, one interior angle is obtuse. The exterior angle adjacent to the obtuse interior angle is acute, and the exterior angles adjacent to the two acute interior angles are obtuse. So, an obtuse triangle always has only one vertex with an acute exterior angle. This statement is True.

Answer:

A right triangle always has obtuse exterior angles at two vertices: True
The sum of the measures of exterior angles minus the sum of the measures of interior angles is 180°: True
The sum of an exterior angle and the two nonadjacent interior angles is 180°: False
An obtuse triangle always has only one vertex with an acute exterior angle: True