Question

in the figure below, suppose ( mangle 2 = 45^{circ} ) and ( mangle 4 = 34^{circ} ).
complete the statements below.
the sum of the interior angle measures of a triangle must be ( square^{circ} ).
so, ( mangle 1 + mangle 2 + mangle 4 = square^{circ} ).
we are given that ( mangle 2 = 45^{circ} ).
so, ( mangle 1 + mangle 4 = square^{circ} ).
from the figure, we can see that ( mangle 2 + mangle 3 = square^{circ} ).
since ( mangle 2 = 45^{circ} ), it must be that ( mangle 3 = square^{circ} ).
therefore, ( mangle 3 ) select ( mangle 1 + mangle 4 ).
this result is an example of the exterior angle property of triangles.
for any triangle, the measure of an exterior angle select

Explanation:

Step1: Sum of interior angles of a triangle

The sum of the interior angle measures of a triangle is \(180^{\circ}\). So, \(m\angle1 + m\angle2+m\angle4=180^{\circ}\).

Step2: Substitute \(m\angle2 = 45^{\circ}\)

If \(m\angle2 = 45^{\circ}\), then \(m\angle1 + m\angle4=180 - 45=135^{\circ}\).

Step3: Linear - pair of angles

Since \(\angle2\) and \(\angle3\) form a linear - pair, \(m\angle2 + m\angle3=180^{\circ}\).

Step4: Find \(m\angle3\)

Given \(m\angle2 = 45^{\circ}\), then \(m\angle3=180 - 45 = 135^{\circ}\).

Answer:

The sum of the interior angle measures of a triangle must be \(180^{\circ}\). So, \(m\angle1 + m\angle2+m\angle4 = 180^{\circ}\). \(m\angle1 + m\angle4=135^{\circ}\). \(m\angle2 + m\angle3 = 180^{\circ}\). \(m\angle3 = 135^{\circ}\). Therefore, \(m\angle3=m\angle1 + m\angle4\). For any triangle, the measure of an exterior angle is equal to the sum of the measures of the two non - adjacent interior angles.