It seems you uploaded a homework form related to geometric theorems, and it involves questions about parallel lines and angle relationships. Let's break down the two questions shown in the image:

1. For the first question:

   • Statement: "a // b"
   • This represents that lines a and b are parallel.
   • A common reason for stating that two lines are parallel can be due to the Given information (stated as a fact in the problem) or because they correspond to same-side interior angles being supplementary or congruent.
   • So, the correct choice is likely Given or Same side interior angles, depending on the context of the problem.

2. For the second question:

   • Statement: "∠3 and ∠2 are supplementary."
   • Supplementary angles sum up to 180°. This often occurs with angles formed by a transversal crossing two parallel lines, where consecutive interior angles (same-side interior angles) are supplementary.
   • Therefore, the best choice here is likely Same side interior angles.

Let me know if you'd like more details or further clarification on any of these topics.

Here are some additional related questions to expand on these concepts:

1. What is the definition of corresponding angles, and how are they used to prove lines are parallel?
2. Can two angles be supplementary if they are not adjacent? Explain with an example.
3. How does the Transitive Property of Equality apply in geometric proofs involving parallel lines?
4. What is the Converse of the Same-Side Interior Angles Theorem?
5. How can congruent alternate interior angles be used to prove lines are parallel?

Tip: When dealing with parallel lines and angles, always look at how a transversal intersects the lines to determine relationships between angles.