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 non - adjacent interior angles is 180°. an obtuse triangle always has only one vertex with an acute exterior angle.

Explanation:

Step1: Analyze right - triangle exterior angles

In a right - triangle, one angle is 90°. The exterior angle at the right - angle vertex is 90°. The other two interior non - right angles are acute. The exterior angles at these two vertices are obtuse. So, a right - triangle always has obtuse exterior angles at two vertices. This statement is True.

Step2: Recall angle - sum properties

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 statement "The sum of the measures of exterior angles minus the sum of the measures of interior angles is 180°" is True.

Step3: Recall exterior - angle theorem

The exterior angle of a triangle is equal to the sum of the two non - adjacent interior angles. Let the exterior angle be \(E\) and the two non - adjacent interior angles be \(A\) and \(B\). Then \(E = A + B\), and \(E+A + B
eq180^{\circ}\). This statement is False.

Step4: Analyze obtuse - triangle exterior angles

In an obtuse triangle, one angle is obtuse. The exterior angle at the obtuse - angle vertex is acute, and the exterior angles at the other two vertices are obtuse. So, an obtuse triangle always has only one vertex with an acute exterior angle. This statement is True.

Answer:

1. True
2. True
3. False
4. True