Question

given that \\(\triangle abc \cong \triangle def\\), determine the value of \\(x\\) in the following diagram:

\\(x = 60\\)
\\(x = 5\\)
\\(x = 12\\)
\\(x = 250\\)

Explanation:

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<reasoning>

Find the measure of angle B

Using the Triangle Angle Sum Theorem knowledge point
\[

$$\begin{aligned} m\angle B &= 180^\circ - (m\angle A + m\angle C) \\ m\angle B &= 180^\circ - (65^\circ + 55^\circ) \\ m\angle B &= 60^\circ \end{aligned}$$

\]

Relate corresponding angles of congruent triangles

Using the Triangle Congruence Correspondence knowledge point
\[

$$\begin{aligned} \Delta ABC &\cong \Delta DEF \\ \angle E &\cong \angle B \\ m\angle E &= m\angle B = 60^\circ \end{aligned}$$

\]

Solve for x

Using the Triangle Congruence Correspondence knowledge point
\[

$$\begin{aligned} 5x &= 60 \\ x &= 12 \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>(A) \(x = 60\)</mcq-option>
<mcq-option>(B) \(x = 5\)</mcq-option>
<mcq-correct>(C) \(x = 12\)</mcq-correct>
<mcq-option>(D) \(x = 250\)</mcq-option>
</answer>

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Answer:

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<reasoning>

Find the measure of angle B

Using the Triangle Angle Sum Theorem knowledge point
\[

$$\begin{aligned} m\angle B &= 180^\circ - (m\angle A + m\angle C) \\ m\angle B &= 180^\circ - (65^\circ + 55^\circ) \\ m\angle B &= 60^\circ \end{aligned}$$

\]

Relate corresponding angles of congruent triangles

Using the Triangle Congruence Correspondence knowledge point
\[

$$\begin{aligned} \Delta ABC &\cong \Delta DEF \\ \angle E &\cong \angle B \\ m\angle E &= m\angle B = 60^\circ \end{aligned}$$

\]

Solve for x

Using the Triangle Congruence Correspondence knowledge point
\[

$$\begin{aligned} 5x &= 60 \\ x &= 12 \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>(A) \(x = 60\)</mcq-option>
<mcq-option>(B) \(x = 5\)</mcq-option>
<mcq-correct>(C) \(x = 12\)</mcq-correct>
<mcq-option>(D) \(x = 250\)</mcq-option>
</answer>

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"Geometry",
"Triangle Congruence Correspondence"
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