Based on the diagram you provided, this is a geometric proof involving parallel lines and congruent angles. The task involves filling out the reasons for the statements provided.

Let's break it down step by step:

Step 3: $\angle SUV \cong \angle RQS$

• Reason: This is due to the fact that $\overline{PR} \parallel \overline{TV}$, and these two angles are corresponding angles formed by the transversal $\overline{QS}$. According to the Corresponding Angles Postulate, if two parallel lines are cut by a transversal, the corresponding angles are congruent.

Step 4: $m\angle RQS + m\angle YXZ = 180^\circ$

• Reason: These two angles are supplementary because they form a linear pair. A linear pair is two adjacent angles whose non-common sides form a straight line, and therefore their measures sum up to 180°. This is by the Linear Pair Postulate.

Step 5: $m\angle SUV + m\angle YXZ = 180^\circ$

• Reason: Similar to step 4, this is another application of the Linear Pair Postulate. These two angles form a linear pair, meaning their measures sum to 180°.

Summary of the Completed Proof:

1. $\overline{PR} \parallel \overline{TV}$ (Given)
2. $\overline{PR} \parallel \overline{WY}$ (Given)
3. $\angle SUV \cong \angle RQS$ (Corresponding Angles Postulate)
4. $m\angle RQS + m\angle YXZ = 180^\circ$ (Linear Pair Postulate)
5. $m\angle SUV + m\angle YXZ = 180^\circ$ (Linear Pair Postulate)

Would you like more details on any of these steps or theorems? Let me know if you have any other questions!

5 Related Questions:

1. What is the Corresponding Angles Postulate?
2. How do you identify supplementary angles in different geometric figures?
3. What is the Linear Pair Postulate, and when can it be applied?
4. How do you know when two angles are congruent in geometric proofs?
5. How does the use of parallel lines simplify geometric proofs?

Tip:

When working with geometric proofs, always look for relationships like parallel lines, transversals, and angle pairs (e.g., corresponding, alternate interior, or linear pairs). These clues will guide you to the appropriate theorems or postulates for proving congruence or supplementary relationships.