Sure, let's solve for the measure of angle [tex]\(a\)[/tex] step-by-step based on the given formula.
Given:
[tex]\[
\angle a = \frac{\hat{c} - \hat{b}}{2}
\][/tex]
We're provided with the measures of two arcs, [tex]\(\hat{c}\)[/tex] and [tex]\(\hat{b}\)[/tex]:
- The measure of arc [tex]\(\hat{c}\)[/tex] is 120 degrees.
- The measure of arc [tex]\(\hat{b}\)[/tex] is 80 degrees.
To find the measure of angle [tex]\(a\)[/tex]:
1. Subtract the measure of arc [tex]\(\hat{b}\)[/tex] from the measure of arc [tex]\(\hat{c}\)[/tex]:
[tex]\[
\hat{c} - \hat{b} = 120 - 80
\][/tex]
This gives us:
[tex]\[
120 - 80 = 40
\][/tex]
2. Divide the result by 2 to find the measure of angle [tex]\(a\)[/tex]:
[tex]\[
\angle a = \frac{40}{2}
\][/tex]
This gives us:
[tex]\[
\angle a = 20
\][/tex]
Therefore, the measure of angle [tex]\(a\)[/tex] is [tex]\(20\)[/tex] degrees.
