Let's go through the provided statements and see which ones are correct based on the diagram:

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

• Incorrect: $\angle 3$ and $\angle 2$ are not alternate interior angles. Instead, they are corresponding angles.

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

• Correct: $\angle 1$ and $\angle 3$ are adjacent angles on a straight line (linear pair), so their measures add up to $180^\circ$.

3. Statement C: $\angle 3 \cong \angle 6$ because they are alternate interior angles.

• Correct: $\angle 3$ and $\angle 6$ are alternate interior angles, formed by the transversal cutting the two parallel lines.

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

• Incorrect: $\angle 1$ and $\angle 6$ are not supplementary. $\angle 1$ is supplementary to $\angle 3$, but this does not imply that $\angle 1$ and $\angle 6$ are supplementary.

5. Statement E: $\angle 1 \cong \angle 3$ because they are vertical angles.

• Incorrect: $\angle 1$ and $\angle 3$ are not vertical angles; they are a linear pair (adjacent on a straight line).

Summary of Correct Statements:

• B and C are correct.

Would you like further details on any of these angles or the reasoning behind the other statements?

Here are 5 related questions to explore further:

1. What are the definitions of alternate interior, corresponding, and vertical angles?
2. How can you identify supplementary and complementary angles in a given diagram?
3. What is the relationship between parallel lines and transversals in determining angle congruence?
4. Can angles on a straight line always be considered supplementary?
5. How do alternate exterior angles compare to alternate interior angles?

Tip: Always double-check the relationships between angles by identifying the transversal and verifying if lines are parallel. This helps in recognizing alternate interior, corresponding, and vertical angles.