Let's analyze the image and solve the problem.

The question asks you to select all the true statements related to the given diagram of intersecting lines with labeled angles. Let’s go through each option systematically:

1. Option A:
   Claim: $\angle 3 \cong \angle 2$ because they are alternate interior angles.
   Analysis:

   • Alternate interior angles are formed on opposite sides of the transversal and inside the two parallel lines.
   • However, $\angle 3$ and $\angle 2$ are not alternate interior angles. They are on the same side of the transversal and thus are corresponding angles.
     Conclusion: This statement is false.

2. Option B:
   Claim: $m\angle 1 + m\angle 3 = 180^\circ$ because they form a straight angle.
   Analysis:

   • $\angle 1$ and $\angle 3$ lie on the same straight line, so their measures add up to $180^\circ$.
     Conclusion: This statement is true.

3. Option C:
   Claim: $\angle 3 \cong \angle 6$ because they are alternate interior angles.
   Analysis:

   • $\angle 3$ and $\angle 6$ are on opposite sides of the transversal and inside the parallel lines. This fits the definition of alternate interior angles.
     Conclusion: This statement is true.

4. Option D:
   Claim: $\angle 1$ and $\angle 6$ are supplementary because $\angle 3 \cong \angle 6$ and $m\angle 1 + m\angle 3 = 180^\circ$.
   Analysis:

   • Since $\angle 1$ and $\angle 3$ add to $180^\circ$, and $\angle 3 \cong \angle 6$, it follows that $m\angle 1 + m\angle 6 = 180^\circ$.
     Conclusion: This statement is true.

5. Option E:
   Claim: $\angle 1 \cong \angle 3$ because they are vertical angles.
   Analysis:

   • Vertical angles are formed by two intersecting lines, but $\angle 1$ and $\angle 3$ are adjacent and lie on a straight line, not opposite to each other.
     Conclusion: This statement is false.

Correct Answers:

• B, C, and D

Let me know if you need further clarifications or have questions!

Here are 5 related questions:

1. What are the properties of corresponding angles?
2. How do alternate interior angles behave with parallel lines?
3. Why do supplementary angles add to $180^\circ$?
4. What is the difference between vertical and adjacent angles?
5. How can you determine if two lines are parallel using angle relationships?

Tip: When working with intersecting lines, always look for key relationships: vertical angles, corresponding angles, and alternate interior angles.