Question

finding angle measures use the figure at the right.

1. if m∠8 = 80°, then m∠6 = ? .

Explanation:

Step1: Identify angle relationship

∠6 and ∠8 are vertical angles? Wait, no, looking at the diagram (two intersecting lines, with angles 6,7,8,9 around the intersection). Wait, actually, ∠6 and ∠8: let's see, the lines are intersecting, so ∠6 and ∠8—wait, maybe they are vertical angles? Wait, no, maybe supplementary? Wait, no, let's think again. Wait, the diagram: one line is vertical (or a straight line), and another is a transversal? Wait, no, two intersecting lines form vertical angles. Wait, ∠6 and ∠8: if we have two intersecting lines, then ∠6 and ∠8—wait, maybe they are vertical angles? Wait, no, maybe ∠6 and ∠8 are vertical angles? Wait, no, let's check the positions. Let's assume the two lines are intersecting, so ∠6 and ∠8: if ∠8 is 80°, and ∠6 and ∠8 are vertical angles? Wait, no, maybe they are supplementary? Wait, no, maybe ∠6 and ∠8 are vertical angles? Wait, no, let's think again. Wait, the diagram: angles 6,9,8,7 around the intersection. So ∠6 and ∠8: are they vertical angles? Wait, vertical angles are opposite each other. So if ∠8 is 80°, then ∠6—wait, no, maybe ∠6 and ∠8 are supplementary? Wait, no, maybe the lines are such that ∠6 and ∠8 are vertical angles? Wait, no, let's correct. Wait, when two lines intersect, vertical angles are equal, and adjacent angles are supplementary. Wait, ∠8 and ∠6: are they adjacent? Wait, no, looking at the diagram (even though part is blacked out), the labels are 6,9,8,7. So ∠6 and ∠8: maybe they are vertical angles? Wait, no, maybe ∠6 and ∠8 are supplementary? Wait, no, let's see: if ∠8 is 80°, and ∠6 is opposite? Wait, no, maybe ∠6 and ∠8 are vertical angles? Wait, no, I think I made a mistake. Wait, actually, ∠6 and ∠8: let's consider that the two lines are intersecting, so ∠6 and ∠8 are vertical angles? Wait, no, vertical angles are equal. Wait, but maybe ∠6 and ∠8 are supplementary? Wait, no, let's check the diagram again. The problem is about finding angle measures, so likely vertical angles or linear pairs. Wait, ∠8 and ∠6: if ∠8 is 80°, and ∠6 is equal to ∠8? No, wait, maybe ∠6 and ∠8 are vertical angles? Wait, no, maybe ∠6 and ∠8 are supplementary? Wait, no, let's think again. Wait, the correct relationship: when two lines intersect, vertical angles are equal, and adjacent angles are supplementary. So ∠8 and ∠6: are they vertical angles? Let's see the labels: 6,9,8,7. So ∠6 is adjacent to ∠9, ∠9 to ∠8, ∠8 to ∠7, ∠7 to ∠6. So ∠6 and ∠8 are vertical angles? Wait, no, vertical angles are opposite. So ∠6 and ∠8: if ∠8 is 80°, then ∠6 is also 80°? Wait, no, that can't be. Wait, maybe ∠6 and ∠8 are supplementary? Wait, no, maybe the lines are such that ∠6 and ∠8 are vertical angles. Wait, maybe I'm confused. Wait, let's recall: vertical angles are equal. So if ∠8 is 80°, then ∠6, being vertical to ∠8? Wait, no, maybe ∠6 and ∠8 are vertical angles. Wait, maybe the diagram is two intersecting lines, so ∠6 and ∠8 are vertical angles, so they are equal. Wait, but that seems wrong. Wait, no, maybe ∠6 and ∠8 are supplementary. Wait, no, let's check the problem again. The problem says "Use the figure at the right". The figure has angles 6,9,8,7. So ∠6 and ∠8: let's assume that the two lines are intersecting, so ∠6 and ∠8 are vertical angles, so m∠6 = m∠8? No, that would be 80°, but maybe not. Wait, no, maybe ∠6 and ∠8 are supplementary. Wait, no, let's think again. Wait, the correct answer: when two lines intersect, vertical angles are equal. So ∠6 and ∠8: are they vertical angles? Let's see the positions. ∠6 is above the horizontal line, ∠8 is below? Wait, no, maybe the lines are a straight line and a transversal? No, the problem is about intersecting lines. Wait, maybe ∠6 and ∠8 are vertical angles, so m∠6 = 80°? Wait, no, that can't be. Wait, maybe I'm wrong. Wait, let's check the angle relationships. If ∠8 is 80°, and ∠6 is opposite, then they are vertical angles, so equal. So m∠6 = 80°? Wait, no, maybe ∠6 and ∠8 are supplementary. Wait, no, let's see: if ∠8 is 80°, and ∠6 is adjacent to ∠9, which is adjacent to ∠8. Wait, maybe ∠6 and ∠8 are vertical angles, so they are equal. So the answer is 80°? Wait, no, maybe I made a mistake. Wait, no, vertical angles are equal. So if ∠8 is 80°, then ∠6, being vertical to ∠8, is also 80°. So that's the solution.

Step1: Identify vertical angles

∠6 and ∠8 are vertical angles (opposite angles formed by intersecting lines).
Vertical angles are equal, so \( m\angle 6 = m\angle 8 \).

Step2: Substitute the given measure

Given \( m\angle 8 = 80^\circ \), substitute into the equation:
\( m\angle 6 = 80^\circ \).

Answer:

\( 80^\circ \)