Question

given: \\(\overline{bc} \cong \overline{ad}\\) and \\(\overline{ab} \cong \overline{cd}\\). prove: \\(\triangle abc \cong \triangle cda\\). note: quadrilateral properties are not permitted in this proof. step 1 statement: \\(\overline{bc} \cong \overline{ad}\\), \\(\overline{ab} \cong \overline{cd}\\) reason: given

Explanation:

Step1: Identify Common Side

In $\triangle ABC$ and $\triangle CDA$, the side $\overline{AC}$ is common to both triangles. So, $\overline{AC} \cong \overline{AC}$ (by the Reflexive Property of Congruence).

Step2: Apply SSS Congruence

We know:

• $\overline{BC} \cong \overline{AD}$ (Given)
• $\overline{AB} \cong \overline{CD}$ (Given)
• $\overline{AC} \cong \overline{AC}$ (Reflexive Property)

By the Side - Side - Side (SSS) Congruence Postulate, if three sides of one triangle are congruent to three sides of another triangle, then the two triangles are congruent. So, $\triangle ABC \cong \triangle CDA$ (by SSS).

Answer:

To prove $\triangle ABC \cong \triangle CDA$:

1. Step 2 Statement: $\overline{AC} \cong \overline{AC}$; Reason: Reflexive Property of Congruence.
2. Step 3 Statement: $\triangle ABC \cong \triangle CDA$; Reason: SSS (Side - Side - Side) Congruence Postulate (since $\overline{BC} \cong \overline{AD}$, $\overline{AB} \cong \overline{CD}$, and $\overline{AC} \cong \overline{AC}$).