Question

Two parallel lines are cut by a transversal and form a pair of alternate exterior angles. One angle measures (6x + 5)° and the other measures (7x – 4)°. Explain how to determine what those angles actually measure?

Answer 1

To determine the measure of the alternate exterior angles formed by the two parallel lines cut by a transversal, we can use the property that alternate exterior angles are equal when the lines are parallel.

1. **Set Up the Equation**: Since the two angles are alternate exterior angles, we can set their measures equal to each other:
\[
6x + 5 = 7x - 4
\]

2. **Solve for x**:
- First, we rearrange the equation to isolate \(x\). Subtract \(6x\) from both sides:
\[
5 = 7x - 6x - 4
\]
- This simplifies to:
\[
5 = x - 4
\]
- Next, add 4 to both sides:
\[
5 + 4 = x
\]
\[
x = 9
\]

3. **Substitute x Back into the Angle Expressions**:
- Now that we have found \(x\), we can substitute it back into the expressions for the angles to find their measures.
- For the first angle:
\[
6x + 5 = 6(9) + 5 = 54 + 5 = 59°
\]
- For the second angle:
\[
7x - 4 = 7(9) - 4 = 63 - 4 = 59°
\]

4. **Conclusion**: Both angles measure \(59°\).

Since these are alternate exterior angles, they remain equal regardless of how you substitute \(x\). Therefore, both angles measure \(59°\).