Let's tackle the questions step-by-step.

3a. Identifying Parallel Lines

The diagram shows two transversals cutting across four lines labeled $a$, $b$, $c$, and $d$. We need to identify which lines are parallel if angles $\angle 3$ and $\angle 5$ are congruent.

In the context of transversals and parallel lines:

• Alternate Interior Angles are congruent when the lines are parallel.

Since $\angle 3$ and $\angle 5$ are alternate interior angles, lines $a$ and $c$ must be parallel.

Thus, the correct answer is:

• C. $a \parallel c$

3b. Justifying the Answer

To justify why $a$ and $c$ are parallel, we must reference the appropriate theorem:

• Alternate Interior Angles Theorem states that if two lines are parallel, then alternate interior angles are congruent. Conversely, if alternate interior angles are congruent, then the lines are parallel.

Therefore, the theorem that justifies this is:

• C. Alternate interior angles of parallel lines cut by a transversal are congruent.

Final Answers:

1. For 3a: C. $a \parallel c$
2. For 3b: C. Alternate interior angles of parallel lines cut by a transversal are congruent.

Would you like further details on the reasoning, or do you have any questions?

Related Questions:

1. What are alternate exterior angles, and when are they congruent?
2. How can we determine if two lines are parallel using corresponding angles?
3. What is the difference between alternate interior angles and consecutive interior angles?
4. How can the Same-Side Interior Angles Theorem help determine parallel lines?
5. What are the criteria for lines to be parallel in terms of angles formed by a transversal?

Tip: When identifying parallel lines using angles, always look for corresponding, alternate interior, or alternate exterior angles, as these are congruent in parallel line scenarios.