Question

in the figure below, suppose ( mangle 1 = 37^circ ) and ( mangle 4 = 108^circ ).
complete the statements below.
the sum of the interior angle measures of a triangle must be ( square^circ ).
so, ( mangle 1 + mangle 3 + mangle 4 = square^circ ).
we are given that ( mangle 1 = 37^circ ) and ( mangle 4 = 108^circ ).
therefore, ( mangle 1 + mangle 4 = square^circ ).
and so ( mangle 3 = square^circ ).
from the figure, we can see that ( mangle 2 + mangle 3 = square^circ ).
using the value we already found for ( mangle 3 ), we find that ( mangle 2 = square^circ ).
therefore, ( mangle 2 ) 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

Explanation:

Step1: Recall triangle angle sum property

The sum of the interior angle measures of a triangle is \(180^\circ\). So, \(m\angle1 + m\angle3 + m\angle4 = 180^\circ\).

Step2: Substitute known angles

We know \(m\angle1 = 37^\circ\) and \(m\angle4 = 108^\circ\). Substitute these into the equation: \(37^\circ + m\angle3 + 108^\circ = 180^\circ\).

Step3: Solve for \(m\angle3\)

First, add \(37^\circ\) and \(108^\circ\): \(37^\circ+ 108^\circ=145^\circ\). Then, \(m\angle3 = 180^\circ - 145^\circ = 35^\circ\).

Step4: Use exterior angle property

The exterior angle property states that an exterior angle of a triangle is equal to the sum of the two non - adjacent interior angles. So, \(m\angle2=m\angle1 + m\angle4\).

Step5: Calculate \(m\angle2\)

Substitute \(m\angle1 = 37^\circ\) and \(m\angle4 = 108^\circ\) into the formula: \(m\angle2=37^\circ + 108^\circ = 145^\circ\). Also, from the straight - line (supplementary angles) property, \(m\angle2 + m\angle3=180^\circ\), and since \(m\angle3 = 35^\circ\), \(m\angle2=180^\circ - 35^\circ = 145^\circ\) (which is consistent with the exterior angle property result).

Answer:

s:

• The sum of the interior angle measures of a triangle must be \(\boldsymbol{180}\)°.
• \(m\angle1 + m\angle3 + m\angle4=\boldsymbol{180}\)°.
• \(m\angle1 + m\angle4=\boldsymbol{145}\)°.
• \(m\angle3=\boldsymbol{35}\)°.
• \(m\angle2 + m\angle3=\boldsymbol{180}\)°.
• \(m\angle2=\boldsymbol{145}\)°.
• The relationship for \(m\angle2\) is \(m\angle2\boldsymbol{=}m\angle1 + m\angle4\).