Question

what is the correct congruency statement for the following triangles?

\\(\bigcirc \delta cab \cong \delta fde\\) by asa
\\(\bigcirc \delta cab \cong \delta fde\\) by aas
\\(\bigcirc \delta cab \cong \delta dfe\\) by aas
\\(\bigcirc \delta cab \cong \delta dfe\\) by asa

Explanation:

Identify corresponding congruent angles

We observe the angle markings on the two triangles:

• Angle \(A\) in \(\triangle CAB\) has a single arc marking.
• Angle \(F\) in \(\triangle DFE\) has a single arc marking.

Thus, \(\angle A \cong \angle F\).

• Angle \(C\) in \(\triangle CAB\) has a double arc marking.
• Angle \(D\) in \(\triangle DFE\) has a double arc marking.

Thus, \(\angle C \cong \angle D\).

Using the Triangle Congruence Correspondence concept, we match the corresponding vertices:

• Vertex \(C\) corresponds to vertex \(D\).
• Vertex \(A\) corresponds to vertex \(F\).
• Vertex \(B\) corresponds to vertex \(E\).

Therefore, the correct triangle correspondence is:

$$\triangle CAB \cong \triangle DFE$$

Identify the congruence theorem

We analyze the marked parts of the triangles:

• Angle \(\angle C \cong \angle D\) (Angle)
• Side \(AC \cong FD\) (Side, which is the included side between the two marked angles)
• Angle \(\angle A \cong \angle F\) (Angle)

Using the Triangle Congruence criteria, since the congruent side is directly between the two congruent angles, the triangles are congruent by the Angle-Side-Angle (ASA) postulate.

Combining these results, the correct statement is:

$$\triangle CAB \cong \triangle DFE \text{ by ASA}$$

Answer:

• (A) \(\triangle CAB \cong \triangle FDE\) by ASA
• (B) \(\triangle CAB \cong \triangle FDE\) by AAS
• (C) \(\triangle CAB \cong \triangle DFE\) by AAS
• (D) \(\triangle CAB \cong \triangle DFE\) by ASA (Correct answer)