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Two windows in a newly designed building are right triangles, with [tex]$\overline{WX} \cong \overline{YX}$[/tex].

Given: [tex]$\triangle WXZ$[/tex] and [tex][tex]$\triangle YXZ$[/tex][/tex] are right triangles; [tex]$\overline{WX} \cong \overline{YX}$[/tex]

Prove: [tex]$\triangle WXZ \cong \triangle YXZ$[/tex]

Which of the following two-column proofs would be correct?

A.
\begin{tabular}{|c|c|}
\hline
\textbf{Statements} & \textbf{Reasons} \\
\hline
[tex]$\triangle WXZ$[/tex] and [tex]$\triangle YXZ$[/tex] are right triangles. & Given \\
\hline
[tex]$WX = YX$[/tex] & Given \\
\hline
[tex]$XZ \cong XZ$[/tex] & Reflexive Property \\
\hline
[tex]$\triangle WXZ \cong \triangle YXZ$[/tex] & Hypotenuse-Leg (HL) Congruence Theorem \\
\hline
\end{tabular}

Sagot :

GinnyAnswer
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}
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