Question

establishing facts about the interior and exterior angles of a triangle
the sum of the interior angle measures of a triangle must be \\(\square^\circ\\).
so, \\(m\angle 2 + m\angle 3 + m\angle 4 = \square^\circ\\).

we are given that \\(m\angle 3 = 118^\circ\\) and \\(m\angle 4 = 32^\circ\\).
therefore, \\(m\angle 3 + m\angle 4 = \square^\circ\\).
and so \\(m\angle 2 = \square^\circ\\).

from the figure, we can see that \\(m\angle 1 + m\angle 2 = \square^\circ\\).
using the value we already found for \\(m\angle 2\\), we find that \\(m\angle 1 = \square^\circ\\).

therefore, \\(m\angle 1\\) select \\(m\angle 3 + m\angle 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: Recall triangle interior angle sum

The sum of interior angles of a triangle is \( 180^\circ \). So, \( m\angle2 + m\angle3 + m\angle4 = 180^\circ \).

Step2: Calculate \( m\angle3 + m\angle4 \)

Given \( m\angle3 = 118^\circ \) and \( m\angle4 = 32^\circ \), then \( m\angle3 + m\angle4 = 118 + 32 = 150^\circ \).

Step3: Find \( m\angle2 \)

Using \( m\angle2 + (m\angle3 + m\angle4) = 180^\circ \), substitute \( m\angle3 + m\angle4 = 150^\circ \). So, \( m\angle2 = 180 - 150 = 30^\circ \).

Step4: Analyze \( m\angle1 + m\angle2 \)

From the figure (linear pair), \( m\angle1 + m\angle2 = 180^\circ \) (since they form a straight line).

Step5: Find \( m\angle1 \)

Substitute \( m\angle2 = 30^\circ \) into \( m\angle1 + m\angle2 = 180^\circ \). So, \( m\angle1 = 180 - 30 = 150^\circ \).

Step6: Compare \( m\angle1 \) and \( m\angle3 + m\angle4 \)

We found \( m\angle1 = 150^\circ \) and \( m\angle3 + m\angle4 = 150^\circ \), so \( m\angle1 = m\angle3 + m\angle4 \).

Step7: Recall Exterior Angle Property

The Exterior Angle Property of Triangles states that the measure of an exterior angle is equal to the sum of the measures of the two non - adjacent interior angles.

Answer:

The sum of the interior angle measures of a triangle must be \( \boldsymbol{180}^\circ \). So, \( m\angle2 + m\angle3 + m\angle4=\boldsymbol{180}^\circ \).

\( m\angle3 + m\angle4 = \boldsymbol{150}^\circ \).

\( m\angle2=\boldsymbol{30}^\circ \).

\( m\angle1 + m\angle2=\boldsymbol{180}^\circ \).

\( m\angle1=\boldsymbol{150}^\circ \).

\( m\angle1 \) \( \boldsymbol{=} \) \( m\angle3 + m\angle4 \).

For any triangle, the measure of an exterior angle \( \boldsymbol{\text{is equal to the sum of the measures of the two non - adjacent interior angles}} \).