The diagram appears to show two intersecting lines, creating angles around point $Z$. We are asked to fill in missing reasons for a proof, using the provided choices. Here’s a breakdown of the important geometrical relationships:

Possible Proof Strategy:

1. Linear Pairs: Angles that are adjacent on a straight line, such as $\angle WZV$ and $\angle XZY$, should form a linear pair, meaning their sum is $180^\circ$.

2. Congruency of Supplementary Angles: If two angles are supplementary to the same angle, then those two angles must be congruent.

3. Right Angles: Perpendicular lines, like the lines shown, form right angles ($90^\circ$).

Reasons Selection:

• "The sum of the measures of a linear pair is 180°" can be used to explain the supplementary nature of the adjacent angles.
• "Definition of Linear Pair" could describe the relationship of two adjacent angles forming a straight line.
• "Angle Addition Postulate" could help describe how two smaller angles add up to a larger angle.
• "Definition of Perpendicular Lines" explains why certain angles are right angles.
• "If two angles are supplementary to the same angle, then the two angles are congruent" would apply in the case where different angles share the same supplementary angle.

Would you like a more specific breakdown of the steps for solving this proof based on the given diagram? Here are some follow-up questions you may consider:

1. What specific angle pairs are supplementary in the diagram?
2. How do perpendicular lines define certain angles in this figure?
3. How does the Angle Addition Postulate apply to this diagram?
4. Can we identify congruent angles based on supplementary relationships?
5. How do we use the definition of linear pairs in a proof?

Tip:

Always start by identifying key angle relationships (linear pairs, right angles) before moving on to the congruency or supplementary conditions.