Let's analyze the problem based on the image provided.

You have two parallel lines $n$ and $m$, cut by a transversal $\ell$. There are angles labeled $\angle 1$, $\angle 2$, and $\angle 3$ at points $R$, $P$, and $Q$ respectively.

The questions ask you to identify the types of angles and explain why based on angle relationships.

Part A:

• $\angle 1$ and $\angle 2$: These angles are on opposite sides of the transversal, and the lines $n$ and $\ell$ are parallel. Angles on opposite sides of the transversal and outside the parallel lines are called alternate exterior angles.
• So, $\angle 1$ and $\angle 2$ are alternate exterior angles and they are congruent because alternate exterior angles formed by a transversal cutting through parallel lines are always congruent.

Part B:

• $\angle 1$ and $\angle 3$: These angles are on the same side of the transversal and outside the parallel lines $n$ and $m$. These are called corresponding angles.
• Therefore, $\angle 1$ and $\angle 3$ are corresponding angles, and they are congruent because corresponding angles formed by a transversal cutting through parallel lines are congruent.

To summarize:

• Part A: $\angle 1$ and $\angle 2$ are alternate exterior angles and they are congruent.
• Part B: $\angle 1$ and $\angle 3$ are corresponding angles and they are congruent.

Would you like to dive into more details on angle relationships or explore other related concepts? Here are some follow-up questions for deeper understanding:

1. What are alternate interior angles, and how are they different from alternate exterior angles?
2. How can we prove that alternate exterior angles are congruent?
3. Are there any cases where corresponding angles are not congruent?
4. What are consecutive (same side) interior angles, and how do they relate to parallel lines?
5. How can you identify a transversal in a given geometric diagram?

Tip: When dealing with parallel lines cut by a transversal, always check the relative position of the angles to classify them correctly (alternate interior, alternate exterior, corresponding, etc.).