First, let's review the given information and understand the setup for proving that the triangles are congruent.
Given:
1. [tex]$\triangle W X Z$[/tex] and [tex]$\triangle Y X Z$[/tex] are right triangles.
2. [tex]$\overline{W X} \cong \overline{Y X}$[/tex].
We need to prove:
- [tex]$\triangle W X Z \cong \triangle Y X Z$[/tex].
To prove that two right triangles are congruent, we can use several criteria such as:
- Hypotenuse-Leg (HL) Congruence Theorem: Two right triangles are congruent if their hypotenuses and one pair of corresponding legs are congruent.
Let's follow through the steps in a possible two-column proof:
(proof segments are described in natural language for ease of understanding)
1. Statement: [tex]$\triangle W X Z$[/tex] and [tex]$\triangle Y X Z$[/tex] are right triangles.
Reason: Given.
2. Statement: [tex]$\overline{W X} \cong \overline{Y X}$[/tex].
Reason: Given.
3. Statement: [tex]$\overline{X Z} \cong \overline{X Z}$[/tex].
Reason: Reflexive Property (a segment is always congruent to itself).
4. Statement: [tex]$\angle W X Z \cong \angle Y X Z = 90^\circ$[/tex] (Right Angles).
Reason: Given (both triangles are right triangles).
Now, we have sufficient information to use the Hypotenuse-Leg (HL) Congruence Theorem for proving the triangles congruent. We know:
- One leg [tex]$\overline{W X} \cong \overline{Y X}$[/tex].
- The hypotenuses [tex]$\overline{X Z} \cong \overline{X Z}$[/tex] (by the Reflexive Property).
- Both triangles have right angles at [tex]$\angle W X Z$[/tex] and [tex]$\angle Y X Z$[/tex].
From this, we conclude:
5. Statement: [tex]$\triangle W X Z \cong \triangle Y X Z$[/tex].
Reason: Hypotenuse-Leg (HL) Congruence Theorem.
Let's present this as a final two-column proof:
\begin{tabular}{|c|c|}
\hline Statements & Reasons \\
\hline
1. [tex]$\triangle W X Z$[/tex] and [tex]$\triangle Y X Z$[/tex] are right triangles & Given \\
2. [tex]$\overline{W X} \cong \overline{Y X}$[/tex] & Given \\
3. [tex]$\overline{X Z} \cong \overline{X Z}$[/tex] & Reflexive Property \\
4. [tex]$\angle W X Z \cong \angle Y X Z = 90^\circ$[/tex] & Given \\
5. [tex]$\triangle W X Z \cong \triangle Y X Z$[/tex] & Hypotenuse-Leg (HL) Congruence Theorem \\
\hline
\end{tabular}
