Question

find the measure of angle 5.
label optional

Explanation:

Step1: Identify Parallel Lines and Transversal

The two slanted lines are parallel (indicated by arrow marks), and the horizontal line is a transversal. So, alternate exterior angles \(2x + 77\) and \(6x + 15\) are equal.
\[2x + 77 = 6x + 15\]

Step2: Solve for \(x\)

Subtract \(2x\) from both sides:
\[77 = 4x + 15\]
Subtract 15 from both sides:
\[62 = 4x\]
Divide by 4:
\[x=\frac{62}{4}=15.5\] Wait, no, correction: \(77 - 15 = 4x\) → \(62 = 4x\)? Wait, \(77 - 15 = 62\), but \(62\div4 = 15.5\)? Wait, maybe I made a mistake. Wait, \(2x + 77 = 6x + 15\) → \(77 - 15 = 6x - 2x\) → \(62 = 4x\)? No, \(77 - 15 = 62\), \(6x - 2x = 4x\), so \(x = 62\div4 = 15.5\)? Wait, but maybe angle 2 and angle 4 are vertical? Wait, no, the two slanted lines are parallel, so the angles \(2x + 77\) (angle 2) and \(6x + 15\) (angle 8's adjacent? Wait, no, angle 2 and angle 6x + 15: wait, maybe they are alternate exterior angles. Wait, let's re - examine the diagram. The horizontal line is a transversal cutting two parallel lines. So angle \(2x + 77\) (angle 2) and angle \(6x + 15\) (angle 8's adjacent? Wait, angle 4 and angle 8 are corresponding angles? Wait, maybe I messed up the angle relationships. Wait, actually, angle \(2x + 77\) (angle 2) and angle \(6x + 15\) (angle 4's corresponding? No, let's see: the two slanted lines are parallel, so the angle \(2x + 77\) (angle 2) and angle \(6x + 15\) (angle 8) are alternate exterior angles? Wait, no, angle 2 and angle 4 are vertical angles? Wait, angle 2 and angle 3 are supplementary, angle 3 and angle 4 are supplementary? No, maybe the correct relationship is that \(2x + 77\) and \(6x + 15\) are equal because they are alternate exterior angles (since lines are parallel, transversal cuts them, so alternate exterior angles are congruent). So:

\(2x + 77 = 6x + 15\)

Subtract \(2x\) from both sides: \(77 = 4x + 15\)

Subtract 15: \(62 = 4x\) → \(x = 15.5\)? Wait, that seems odd. Wait, maybe I got the angles wrong. Wait, angle 5 and angle 4: are they supplementary? Wait, angle 4 is \(6x + 15\), angle 5 is adjacent to angle 4, so angle 4 + angle 5 = 180 (linear pair). Also, angle \(2x + 77\) is equal to angle 4 (vertical angles? Because angle 2 and angle 4 are vertical angles). Oh! That's the mistake. Angle 2 and angle 4 are vertical angles, so they are equal. And angle 4 and angle 6x + 15: wait, no, the two slanted lines are parallel, so angle 4 and angle 8 are corresponding angles, and angle 8 and angle 6x + 15 are supplementary? No, let's start over.

1. Angle 2 (\(2x + 77\)) and angle 4 are vertical angles, so \(m\angle2=m\angle4\).
2. The two slanted lines are parallel, so angle 4 and angle \(6x + 15\) are equal (corresponding angles).
3. Therefore, \(2x + 77=6x + 15\)

Now, solve for \(x\):

Subtract \(2x\) from both sides: \(77 = 4x+15\)

Subtract 15 from both sides: \(62 = 4x\) → \(x=\frac{62}{4}=15.5\)? Wait, that can't be. Wait, maybe angle 2 and angle \(6x + 15\) are supplementary? Wait, if the lines are parallel, and angle 2 and angle \(6x + 15\) are same - side exterior angles, then they are supplementary. Let's try that.

If they are same - side exterior angles, then \(2x + 77+6x + 15 = 180\)

Combine like terms: \(8x+92 = 180\)

Subtract 92: \(8x=180 - 92=88\)

Divide by 8: \(x = 11\)

Ah, this makes more sense. I must have misidentified the angle relationship. Same - side exterior angles are supplementary when lines are parallel.

So \(2x + 77+6x + 15 = 180\)

\(8x+92 = 180\)

\(8x=180 - 92 = 88\)

\(x = 11\)

Step3: Find the measure of angle 4 (or angle 2)

Substitute \(x = 11\) into \(2x + 77\):

\(2\times11+77=22 + 77=99\)

So \(m\angle4 = 99^{\circ}\)

Step4: Find the measure of angle 5

Angle 4 and angle 5 are a linear pair, so they are supplementary.

\(m\angle5=180 - m\angle4\)

\(m\angle5 = 180-99 = 81^{\circ}\)

Answer:

\(81^{\circ}\)