Question

state what additional information is needed to prove the triangles are congruent using sas:

\\(hj \cong rt\\)
\\(hi \cong st\\)
\\(ij \cong st\\)
not possible

Explanation:

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<reasoning>

Identify the given congruent parts

Using the Triangle Congruence Postulates and SAS Congruence knowledge points
\[

$$\begin{aligned} &\angle I \cong \angle S \quad \text{(indicated by angle arcs)}\\ &IH \cong SR \quad \text{(indicated by tick marks)} \end{aligned}$$

\]

Determine the required side for SAS

Using the SAS Congruence knowledge point
\[

$$\begin{aligned} &\text{SAS requires the congruent angle to be included between two congruent sides.}\\ &\text{For }\triangle JIH\text{, the sides forming }\angle I\text{ are }JI\text{ and }IH.\\ &\text{For }\triangle TSR\text{, the sides forming }\angle S\text{ are }TS\text{ and }SR.\\ &\text{Since we have }IH \cong SR\text{ and }\angle I \cong \angle S\text{, we need }IJ \cong ST. \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>(A) \(HJ \cong RT\)</mcq-option>
<mcq-option>(B) \(HI \cong ST\)</mcq-option>
<mcq-correct>(C) \(IJ \cong ST\)</mcq-correct>
<mcq-option>(D) not possible</mcq-option>
</answer>

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Answer:

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<reasoning>

Identify the given congruent parts

Using the Triangle Congruence Postulates and SAS Congruence knowledge points
\[

$$\begin{aligned} &\angle I \cong \angle S \quad \text{(indicated by angle arcs)}\\ &IH \cong SR \quad \text{(indicated by tick marks)} \end{aligned}$$

\]

Determine the required side for SAS

Using the SAS Congruence knowledge point
\[

$$\begin{aligned} &\text{SAS requires the congruent angle to be included between two congruent sides.}\\ &\text{For }\triangle JIH\text{, the sides forming }\angle I\text{ are }JI\text{ and }IH.\\ &\text{For }\triangle TSR\text{, the sides forming }\angle S\text{ are }TS\text{ and }SR.\\ &\text{Since we have }IH \cong SR\text{ and }\angle I \cong \angle S\text{, we need }IJ \cong ST. \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>(A) \(HJ \cong RT\)</mcq-option>
<mcq-option>(B) \(HI \cong ST\)</mcq-option>
<mcq-correct>(C) \(IJ \cong ST\)</mcq-correct>
<mcq-option>(D) not possible</mcq-option>
</answer>

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