To determine the measures of the alternate exterior angles formed by two parallel lines cut by a transversal, follow these steps:
1. Set up the Equation:
- Since the lines are parallel, the alternate exterior angles are equal.
- Therefore, we can write the equation:
[tex]\[
6x + 5 = 7x - 4
\][/tex]
2. Solve for [tex]\(x\)[/tex]:
- First, simplify the equation by getting all terms involving [tex]\(x\)[/tex] on one side and constants on the other:
[tex]\[
6x + 5 = 7x - 4
\][/tex]
Subtract [tex]\(6x\)[/tex] from both sides:
[tex]\[
5 = x - 4
\][/tex]
Add 4 to both sides to isolate [tex]\(x\)[/tex]:
[tex]\[
5 + 4 = x
\][/tex]
Simplify:
[tex]\[
x = 9
\][/tex]
3. Substitute [tex]\(x\)[/tex] back into one of the angle expressions:
- Now that we have [tex]\(x = 9\)[/tex], we can substitute this value back into either of the angle expressions to find the actual measure of the angles.
- Using the first angle expression [tex]\((6x + 5)^\circ\)[/tex]:
[tex]\[
6(9) + 5 = 54 + 5 = 59^\circ
\][/tex]
4. Verify using the second angle expression:
- To ensure consistency, we can check the second angle expression [tex]\((7x - 4)^\circ\)[/tex]:
[tex]\[
7(9) - 4 = 63 - 4 = 59^\circ
\][/tex]
- Since both expressions yield the same angle measure, we have verified our solution.
Thus, the measure of each alternate exterior angle is [tex]\(59^\circ\)[/tex].
