Sure, let's solve this step-by-step.
Given:
- The sum of the measures of all three angles in a triangle is [tex]\(180\)[/tex] degrees.
- The sum of the two base angles is [tex]\(77.5\)[/tex] degrees.
To find the measure of the vertex angle, follow these steps:
1. Understanding the sum of angles in a triangle:
In any triangle, the sum of the three interior angles is always [tex]\(180\)[/tex] degrees.
2. Determine the sum of the base angles:
We are given that the sum of the two base angles is [tex]\(77.5\)[/tex] degrees.
3. Calculate the vertex angle:
The vertex angle is the angle that completes the total [tex]\(180\)[/tex] degrees when added to the sum of the base angles. Therefore, its measure can be calculated as follows:
[tex]\[
\text{Vertex Angle} = 180^\circ - \text{Sum of Base Angles}
\][/tex]
4. Substitute the given value:
Substitute [tex]\(77.5\)[/tex] degrees for the sum of the base angles:
[tex]\[
\text{Vertex Angle} = 180^\circ - 77.5^\circ
\][/tex]
5. Perform the subtraction:
[tex]\[
\text{Vertex Angle} = 102.5^\circ
\][/tex]
So, the measure of the vertex angle is [tex]\(102.5\)[/tex] degrees.
