Let's begin by looking at the given information and the desired outcome of the proof. Then, we can write a paragraph proof!
Given:& Y is the midpoint of XZ and & Z is the midpoint of YW
Prove:& XY ≅ ZW
We know that Y is the midpoint of XZ. Therefore, by the definition of a midpoint, X Y= Y Z. Similarly, knowing that Z is the midpoint of YW, we can conclude that Y Z= Z W.
\begin{gathered}
\underline\textbf{Statement}\\
\text{If } Y \text{ is the midpoint of } \overline{XZ} \text{ and } \\ Z \text{ is the midpoint of } \overline{YW}, \\ \text{then from the definition of a midpoint} \\ \text{of a segment } XY=YZ \text{ and } YZ=ZW.
\end{gathered}
Now, notice that we can substitute ZW for YZ in the equation XY=YZ. Then, we will get XY= ZW.
\begin{gathered}
\underline\textbf{Statement}\\
\text{By substitution, } XY=ZW.
\end{gathered}
Finally, by the definition of congruence, if two segments have the same measure, then they are congruent. Therefore, XY ≅ ZW, which is what we wanted to prove.
\begin{gathered}
\underline\textbf{Statement}\\
\text{By the definition of congruence, if two segments} \\ \text{have the same measure, then they are} \\ \text{congruent. Thus, } \overline{XY} \cong \overline{ZW}.
\end{gathered}
Final Proof
Given:& Y is the midpoint of XZ and & Z is the midpoint of YW
Prove:& XY ≅ ZW
Prove: If Y is the midpoint of XZ and Z is the midpoint of YW, then from the definition of a midpoint of a segment XY=YZ and YZ=ZW. By substitution, XY=ZW. By the definition of congruence, if two segments have the same measure, then they are congruent. Thus, XY ≅ ZW.
