Let's go through each of the three problems on the worksheet.

1. Is $AB \parallel CD$?

We are asked to determine whether lines $AB$ and $CD$ are parallel using the given angle measurements.

1. If $m \angle 5 = 30^\circ$ and $m \angle 3 = 150^\circ$:

• Angles 5 and 3 are on opposite sides of the transversal and are alternate interior angles.
• If alternate interior angles are congruent, the lines are parallel.
• But here, $\angle 5 = 30^\circ$ and $\angle 3 = 150^\circ$, which are not congruent.

Answer: No, because angles 5 and 3 are not equal (they are alternate interior angles).

2. If $m \angle 4 = 80^\circ$ and $m \angle 5 = 100^\circ$:

• Angles 4 and 5 are on the same side of the transversal and should be supplementary (add to $180^\circ$) if the lines were parallel, as they are same-side interior angles.
• $80^\circ + 100^\circ = 180^\circ$, so the lines are parallel.

Answer: Yes, because angles 4 and 5 are same-side interior angles, and their measures add up to $180^\circ$.

3. If $m \angle 2 = 140^\circ$ and $m \angle 7 = 140^\circ$:

• Angles 2 and 7 are on opposite sides of the transversal, and they are corresponding angles.
• Corresponding angles are congruent when lines are parallel.
• Since $m \angle 2 = m \angle 7 = 140^\circ$, the lines are parallel.

Answer: Yes, because angles 2 and 7 are corresponding angles and congruent.

Do you have any questions or want details on any of these steps?

Here are 5 related questions:

1. What are alternate interior angles, and how do they help in determining parallel lines?
2. How can we determine whether two lines are parallel using same-side interior angles?
3. What are corresponding angles, and how do they indicate that two lines are parallel?
4. If two angles are not congruent, can the lines still be parallel? Why or why not?
5. How can you use vertical angles to help prove lines are parallel?

Tip:

Always check if the sum of same-side interior angles equals $180^\circ$ when trying to prove lines are parallel!