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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]
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.
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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