The image shows a geometric proof based on segment addition and properties of equality. The goal is to prove that $PR = QS$, given $PQ = RS$.

Here is a breakdown of the steps involved in completing the proof:

1. Given: $PQ = RS$

   • This is provided directly in the problem statement.

2. Addition Property of Equality:

   • We add the same value $QR$ to both sides of the given equation: $PQ + QR = RS + QR$

3. Segment Addition Postulate (Postulate 1.2):

   • The postulate states that the sum of the lengths of two adjacent segments is the length of the entire segment. $PQ + QR = PR$
   • Similarly: $RS + QR = QS$

4. Transitive Property of Equality:

   • Since both $PR$ and $QS$ are equal to the same expression (i.e., $PQ + QR$ and $RS + QR$), we conclude that: $PR = QS$

This completes the proof.

Would you like further details or any clarifications?

Here are five related questions to deepen your understanding:

1. How does the Segment Addition Postulate work in more complex geometric proofs?
2. Can the Transitive Property of Equality be applied in other types of proofs?
3. What is the difference between the Addition Property of Equality and the Segment Addition Postulate?
4. How could this proof be extended if other segments are introduced?
5. What role does symmetry play in geometric proofs like this one?

Tip: When working on geometry proofs, always start by identifying given information and the postulates or properties that directly apply to the situation. This provides a clear path toward the conclusion.