Solve for the measure of angle [tex]a[/tex].

If two secant lines intersect outside a circle:
[tex]\[ \angle a = \frac{\hat{c} - \hat{b}}{2} \][/tex]

Where:
- [tex]\hat{c}[/tex] and [tex]\hat{b}[/tex] are the measures of the intercepted arcs.



Answer :

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.