The diagram and questions provided seem to focus on the relationship between angles and parallel lines, using transversals to derive conclusions about the angles. Here's how we can approach each problem:

1. If $m \angle 1 = m \angle 8$, then ____ $\parallel$ ____.

• Reason: If corresponding angles are equal, then the two lines are parallel. In this case, angles $1$ and $8$ are corresponding angles. Therefore, the two lines cut by the transversal are $a$ and $b$.
• Answer: $a \parallel b$
• Reason: Corresponding angles theorem.

2. If $m \angle 7 = m \angle 3$, then ____ $\parallel$ ____.

• Reason: Angles $7$ and $3$ are alternate interior angles. When alternate interior angles are congruent, the lines are parallel.
• Answer: $a \parallel b$
• Reason: Alternate interior angles theorem.

3. If $m \angle 4 + m \angle 6 = 180^\circ$, then ____ $\parallel$ ____.

• Reason: Angles $4$ and $6$ are consecutive interior angles (or same-side interior angles). If the sum of consecutive interior angles is 180°, the lines are parallel.
• Answer: $a \parallel b$
• Reason: Consecutive (same-side) interior angles theorem.

4. If $m \angle 5 = m \angle 4$, then ____ $\parallel$ ____.

• Reason: Angles $5$ and $4$ are corresponding angles. If corresponding angles are equal, then the lines are parallel.
• Answer: $a \parallel b$
• Reason: Corresponding angles theorem.

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Related Questions:

1. How can you identify alternate interior angles in any transversal diagram?
2. What are the conditions for consecutive interior angles to imply parallel lines?
3. Why do corresponding angles imply parallelism between lines?
4. Can same-side exterior angles also be used to prove lines are parallel?
5. What other methods can you use to prove lines are parallel without using angle measures?

Tip:

Remember that parallel lines never intersect, and transversals create predictable angle relationships that help in proving parallelism.