Question

show that the measure of an exterior angle is always equal to the sum of the measures of its non - adjacent interior angles. use the figure below to help.
first identify the types of angles in the figure.
the exterior angle is ∠4. the non - adjacent interior angles are ∠1 and ∠3
non - adjacent angles are on the opposite side of the triangle from the exterior angle

Explanation:

Step1: Use angle - sum property of a triangle

In a triangle, the sum of interior angles is \(180^{\circ}\). So, \(\angle1+\angle3+\angle5 = 180^{\circ}\), where \(\angle5\) is the interior angle adjacent to \(\angle2\) (since \(\angle2\) and \(\angle5\) form a linear pair). Then \(\angle5=180^{\circ}-\angle2\). Substituting \(\angle5 = 180^{\circ}-\angle2\) into \(\angle1+\angle3+\angle5 = 180^{\circ}\), we get \(\angle1+\angle3+(180^{\circ}-\angle2)=180^{\circ}\).

Step2: Simplify the equation

Subtract \(180^{\circ}\) from both sides of the equation \(\angle1+\angle3+(180^{\circ}-\angle2)=180^{\circ}\). We have \(\angle1+\angle3-\angle2=0\). Then, by adding \(\angle2\) to both sides, we get \(\angle2=\angle1 + \angle3\).

Answer:

The measure of an exterior angle (\(\angle2\)) of a triangle is equal to the sum of the measures of its non - adjacent interior angles (\(\angle1\) and \(\angle3\)) as shown by using the angle - sum property of a triangle (\(\angle1+\angle3+\angle5 = 180^{\circ}\)) and the linear - pair property (\(\angle2+\angle5 = 180^{\circ}\)).