Question

the measure of angle 1 is 60 degrees. what is the measure of angle 8?

Explanation:

Step1: Identify Vertical Angles

Angle 1 and angle 3 are vertical angles? No, wait, angle 1 and angle 2 are adjacent, angle 1 and angle 3? Wait, actually, angle 1 and angle 3: no, let's look at the transversal. Wait, lines Q and M are parallel? Wait, angle 1 and angle 5: no, wait, angle 1 and angle 2 are supplementary? Wait, no, angle 1 is 60 degrees. Then angle 2 is vertical to angle 4? Wait, maybe first, angle 1 and angle 3: no, angle 1 and angle 2 are adjacent, forming a linear pair? Wait, no, the diagram: two parallel lines (Q and M) cut by a transversal (N). So angle 1 and angle 5: corresponding angles? Wait, no, angle 1 is at the top of line Q, transversal N. Then angle 8: let's see, angle 1 and angle 2 are vertical? Wait, no, angle 1 and angle 3: no, maybe angle 1 and angle 5 are corresponding, but wait, angle 8: let's find the relationship.

Wait, angle 1 and angle 2: vertical angles? No, angle 1 and angle 3: adjacent, forming a linear pair? Wait, angle 1 is 60 degrees, so angle 2 is 180 - 60 = 120? No, wait, maybe angle 1 and angle 3 are vertical? No, the diagram: transversal N intersects Q and M (parallel). So angle 1 and angle 5: corresponding? Wait, angle 5 and angle 8: vertical angles? No, angle 5 and angle 7: vertical, angle 6 and angle 8: vertical. Wait, angle 1 and angle 2: vertical? Wait, maybe I made a mistake. Let's re-examine:

Line Q (vertical arrows) and line M (vertical arrows) are parallel. Transversal N (diagonal) intersects them. So angle 1 is at the top intersection, between transversal N and line Q (top right). Angle 8 is at the bottom intersection, between transversal N and line M (bottom left).

Angle 1 and angle 5: corresponding angles (since Q || M, transversal N). So angle 1 = angle 5 = 60 degrees. Then angle 5 and angle 8: what's their relationship? Angle 5 and angle 8: are they alternate interior angles? Wait, angle 5 is at the top of M, transversal N (left side), angle 8 is at the bottom of M, transversal N (left side). Wait, no, angle 5 and angle 8: vertical angles? No, angle 5 and angle 7 are vertical, angle 6 and angle 8 are vertical. Wait, angle 5 and angle 8: maybe same-side interior? No, angle 5 and angle 8: let's see, angle 5 + angle 6 = 180 (linear pair), angle 6 and angle 8 = vertical, so angle 5 + angle 8 = 180? No, that can't be. Wait, maybe angle 1 and angle 8: let's find the path.

Wait, angle 1 and angle 2: vertical angles? No, angle 1 and angle 3: adjacent, linear pair. Wait, angle 1 is 60, so angle 3 is 180 - 60 = 120? No, maybe angle 1 and angle 2 are vertical, so angle 1 = angle 2 = 60? Wait, no, vertical angles are equal. Wait, angle 1 and angle 3: no, angle 1 and angle 2: if they are vertical, then angle 1 = angle 2 = 60. Then angle 2 and angle 6: corresponding angles (Q || M, transversal N), so angle 2 = angle 6 = 60. Then angle 6 and angle 8: vertical angles, so angle 6 = angle 8 = 60? Wait, that makes sense.

Wait, let's try again:

1. Angle 1 and angle 2: vertical angles (opposite each other at the intersection of Q and N), so angle 1 = angle 2 = 60° (since vertical angles are equal).

2. Line Q || line M (parallel), transversal N. So angle 2 and angle 6: corresponding angles (same position relative to parallel lines and transversal), so angle 2 = angle 6 = 60° (corresponding angles are equal when lines are parallel).

3. Angle 6 and angle 8: vertical angles (opposite each other at the intersection of M and N), so angle 6 = angle 8 = 60° (vertical angles are equal).

So angle 8 is 60 degrees.

Step2: Verify the Relationships

- Vertical Angles: angle 1 = angle 2 (vertical), so 60° = 60°.
- Corresponding Angles: angle 2 = angle 6 (Q || M, transversal N), so 60° = 60°.
- Vertical Angles: angle 6 = angle 8 (vertical), so 60° = 60°.

Thus, angle 8 is 60 degrees.

Answer:

60 degrees