Let's analyze the image and determine the truth of the given statements.

The image depicts two sets of intersecting lines: $p$, $q$, $r$, and $s$, with labeled angles from 1 to 8.

Step-by-step reasoning:

A. $p \parallel q$ if $\angle 2 \cong \angle 3$

• $\angle 2$ and $\angle 3$ are on opposite sides of the transversal (line $r$) and are alternate interior angles.
• If $\angle 2 \cong \angle 3$, then by the alternate interior angles theorem, lines $p$ and $q$ are parallel.
• This statement is true.

B. $p \parallel q$ if $\angle 5 \cong \angle 7$

• $\angle 5$ and $\angle 7$ are on opposite sides of the transversal (line $r$) and are corresponding angles.
• If $\angle 5 \cong \angle 7$, lines $p$ and $q$ are parallel by the corresponding angles postulate.
• This statement is true.

C. $r \parallel s$ if $\angle 2 \cong \angle 4$

• $\angle 2$ and $\angle 4$ are on the same side of the transversal (line $p$) and are consecutive interior angles.
• If $\angle 2 \cong \angle 4$, then by the consecutive interior angles theorem, lines $r$ and $s$ are not parallel, because consecutive interior angles must sum to 180° for the lines to be parallel.
• This statement is false.

D. $r \parallel s$ if $\angle 5 \cong \angle 6$

• $\angle 5$ and $\angle 6$ are on opposite sides of the transversal (line $p$) and are alternate interior angles.
• If $\angle 5 \cong \angle 6$, then by the alternate interior angles theorem, lines $r$ and $s$ are parallel.
• This statement is true.

Conclusion:

• The true statements are A, B, and D.

Let me know if you would like further details or explanations!

Related questions:

1. What are alternate interior angles, and how are they used in proving parallel lines?
2. What is the difference between corresponding and alternate interior angles?
3. Can lines be parallel if consecutive interior angles are congruent?
4. How does the transversal affect the relationships between angles?
5. What postulate is used to determine if corresponding angles imply parallel lines?

Tip: When analyzing parallel lines and transversals, always check if the angles given are alternate interior, corresponding, or consecutive interior angles.