Let's go through the problems in the image.

Question 2:

The diagram shows two parallel lines $a$ and $b$ intersected by a transversal $t$, which creates various angles labeled 1 through 8.

We are asked to determine whether certain statements are True or False:

• (a) $\angle 1 \cong \angle 5$
  Answer: True.
  These are corresponding angles when line $a$ is parallel to line $b$, so they are congruent.

• (b) $m\angle 2 = m\angle 8$
  Answer: True.
  These are alternate exterior angles, and alternate exterior angles are equal when lines are parallel.

• (c) $\angle 3$ and $\angle 6$ are supplementary
  Answer: True.
  These are same-side interior angles (also known as consecutive interior angles), and they are supplementary (add up to $180^\circ$).

• (d) $m\angle 4 + m\angle 7 = 180^\circ$
  Answer: True.
  These are consecutive interior angles formed by the transversal, and they are supplementary.

Question 3:

You are given a diagram with four lines ($a$, $b$, $c$, and $d$) intersected by a transversal. The angles formed by the intersections are labeled with degree measures.

You are asked to determine which lines are parallel based on the given angle measures.

• Line $a$: $115^\circ$
• Line $b$: $120^\circ$
• Line $c$: $55^\circ$
• Line $d$: $65^\circ$

To determine parallelism, you check for corresponding or alternate angles that are equal or supplementary.

• $120^\circ$ (line $b$) and $65^\circ$ (line $d$) are not supplementary (they do not add up to $180^\circ$).
• However, lines $a$ and $c$ have corresponding angles of $115^\circ$ and $55^\circ$, respectively, which do add up to 180°. This means lines $a$ and $c$ must be parallel.

Answer: Line $a$ is parallel to line $c$.

Would you like more details or have any further questions on this?
Here are 5 related questions to consider:

1. How do corresponding angles help identify parallel lines?
2. Can alternate interior angles be used to check parallelism?
3. Why are same-side interior angles always supplementary for parallel lines?
4. How can vertical angles help solve angle congruence problems in parallel lines?
5. What happens if the transversal is not perpendicular to the parallel lines?

Tip: Always remember that corresponding angles are congruent when lines are parallel, and consecutive interior angles are supplementary.