Question

complete the statements below.
the sum of the interior angle measures of a triangle must be \\(\square\\)°.
so, \\(m\angle 1 + m\angle 2 + m\angle 3 = \square\\)°.

we are given that \\(m\angle 3 = 77\\)°.
so, \\(m\angle 1 + m\angle 2 = \square\\)°.

from the figure, we can see that \\(m\angle 3 + m\angle 4 = \square\\)°.
since \\(m\angle 3 = 77\\)°, it must be that \\(m\angle 4 = \square\\)°.

therefore, \\(m\angle 4\\) select \\(m\angle 1 + m\angle 2\\).

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 Angle Sum

The sum of interior angles of a triangle is \(180^\circ\). So, \(m\angle1 + m\angle2 + m\angle3 = 180^\circ\).

Step2: Calculate \(m\angle1 + m\angle2\)

Given \(m\angle3 = 77^\circ\), subtract from \(180^\circ\): \(180 - 77 = 103^\circ\). So, \(m\angle1 + m\angle2 = 103^\circ\).

Step3: Recall Linear Pair Sum

A linear pair (adjacent supplementary angles) sums to \(180^\circ\), so \(m\angle3 + m\angle4 = 180^\circ\).

Step4: Calculate \(m\angle4\)

Substitute \(m\angle3 = 77^\circ\): \(180 - 77 = 103^\circ\). So, \(m\angle4 = 103^\circ\).

Step5: Compare \(m\angle4\) and \(m\angle1 + m\angle2\)

Since both are \(103^\circ\), \(m\angle4 = m\angle1 + m\angle2\).

Answer:

The sum of the interior angle measures of a triangle must be \(\boldsymbol{180}\)°.
So, \(m\angle1 + m\angle2 + m\angle3 = \boldsymbol{180}\)°.
We are given that \(m\angle3 = 77^\circ\).
So, \(m\angle1 + m\angle2 = \boldsymbol{103}\)°.
From the figure, we can see that \(m\angle3 + m\angle4 = \boldsymbol{180}\)°.
Since \(m\angle3 = 77^\circ\), it must be that \(m\angle4 = \boldsymbol{103}\)°.
Therefore, \(m\angle4\) \(\boldsymbol{=}\) \(m\angle1 + m\angle2\).