Question

what additional information do you need to know in order to prove the two triangles congruent using hl?

\\(\angle xyw \cong \angle wyz\\)
\\(\angle xwy \cong \angle zwy\\)
\\(xw \cong yz\\)
no additional information needed

Explanation:

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<reasoning>

Identify given geometric properties

Using the HL Congruence Theorem knowledge point
\[

$$\begin{aligned} &\angle X = 90^\circ, \quad \angle Z = 90^\circ \implies \triangle XYW \text{ and } \triangle ZWY \text{ are right triangles}\\ &XY = WZ \quad (\text{marked legs})\\ &WY = YW \quad (\text{shared hypotenuse by Reflexive Property}) \end{aligned}$$

\]

Determine sufficiency for HL Congruence

Using the HL Congruence Theorem knowledge point
\[

$$\begin{aligned} &\text{Hypotenuse: } WY \cong YW\\ &\text{Leg: } XY \cong ZW\\ &\text{Right Angles: } \angle X \cong \angle Z\\ &\implies \triangle XYW \cong \triangle ZWY \text{ by HL Theorem} \end{aligned}$$

\]

Evaluate multiple-choice options

Using the HL Congruence Theorem knowledge point
\[

$$\begin{aligned} &\text{Since the triangles are already congruent by HL with given markings,}\\ &\text{no additional information is needed.} \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>\(\angle XYW \cong \angle WYZ\)</mcq-option>
<mcq-option>\(\angle XWY \cong \angle ZWY\)</mcq-option>
<mcq-option>\(XW \cong YZ\)</mcq-option>
<mcq-correct>No additional information needed</mcq-correct>
</answer>

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Answer:

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<reasoning>

Identify given geometric properties

Using the HL Congruence Theorem knowledge point
\[

$$\begin{aligned} &\angle X = 90^\circ, \quad \angle Z = 90^\circ \implies \triangle XYW \text{ and } \triangle ZWY \text{ are right triangles}\\ &XY = WZ \quad (\text{marked legs})\\ &WY = YW \quad (\text{shared hypotenuse by Reflexive Property}) \end{aligned}$$

\]

Determine sufficiency for HL Congruence

Using the HL Congruence Theorem knowledge point
\[

$$\begin{aligned} &\text{Hypotenuse: } WY \cong YW\\ &\text{Leg: } XY \cong ZW\\ &\text{Right Angles: } \angle X \cong \angle Z\\ &\implies \triangle XYW \cong \triangle ZWY \text{ by HL Theorem} \end{aligned}$$

\]

Evaluate multiple-choice options

Using the HL Congruence Theorem knowledge point
\[

$$\begin{aligned} &\text{Since the triangles are already congruent by HL with given markings,}\\ &\text{no additional information is needed.} \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>\(\angle XYW \cong \angle WYZ\)</mcq-option>
<mcq-option>\(\angle XWY \cong \angle ZWY\)</mcq-option>
<mcq-option>\(XW \cong YZ\)</mcq-option>
<mcq-correct>No additional information needed</mcq-correct>
</answer>

<post_analysis>
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