Let's tackle the segment proofs shown in your worksheet.

For part (a), you're proving that $XY = \frac{1}{2} XZ$ given that $Y$ is the midpoint of $XZ$. Here's how you can fill out the two-column proof:

1. Statement: $Y$ is the midpoint of $XZ$
   Reason: Given
2. Statement: $XY = YZ$
   Reason: Definition of a midpoint
3. Statement: $XZ = XY + YZ$
   Reason: Segment addition postulate
4. Statement: $XZ = XY + XY$
   Reason: Substitution (since $XY = YZ$)
5. Statement: $XZ = 2XY$
   Reason: Simplification
6. Statement: $XY = \frac{1}{2} XZ$
   Reason: Division property of equality

For part (b), you're proving that $AC \cong BD$ given that $AB \cong CD$. Here's how to fill in the two-column proof:

1. Statement: $AB \cong CD$
   Reason: Given
2. Statement: $AC = AB + BC$
   Reason: Segment addition postulate
3. Statement: $BD = BC + CD$
   Reason: Segment addition postulate
4. Statement: $AB \cong CD$
   Reason: Given (or restate the original)
5. Statement: $AC = BD$
   Reason: Substitution (since $AB \cong CD$)
6. Statement: $AC \cong BD$
   Reason: Definition of congruence

These should help you complete both proofs. Would you like further details on any of these steps?

Here are 5 related questions to expand your understanding:

1. How does the midpoint definition help in proving segment relationships?
2. Can segment addition be applied in other geometric figures, like triangles?
3. How would you prove congruence for angles using a two-column proof?
4. What are some common postulates used in segment congruence proofs?
5. How does the substitution property assist in proofs like these?

Tip:

Always label the diagram with given information before starting your proof. It can help clarify relationships between the parts of a figure!