Question

given: \\(\overline{bc} \cong \overline{ad}\\) and \\(\overline{bc} \parallel \overline{ad}\\).
prove: \\(\triangle abc \cong \triangle cda\\).
step 1
statement: \\(\overline{bc} \cong \overline{ad}\\), \\(\overline{bc} \parallel \overline{ad}\\)
reason: given
type of statement

Explanation:

Step1: Identify Alternate Interior Angles

Since $\overline{BC} \parallel \overline{AD}$ and $\overline{AC}$ is a transversal, $\angle BCA \cong \angle DAC$ (Alternate Interior Angles Theorem).

Step2: Identify Common Side

$\overline{AC} \cong \overline{AC}$ (Reflexive Property of Congruence).

Step3: Apply SAS Congruence Criterion

We have $\overline{BC} \cong \overline{AD}$ (Given), $\angle BCA \cong \angle DAC$ (from Step 1), and $\overline{AC} \cong \overline{AC}$ (from Step 2). By the Side - Angle - Side (SAS) Congruence Postulate, $\triangle ABC \cong \triangle CDA$.

Answer:

We prove $\triangle ABC \cong \triangle CDA$ as follows:

1. $\overline{BC} \cong \overline{AD}$ (Given)
2. $\overline{BC} \parallel \overline{AD}$ (Given)
3. $\angle BCA \cong \angle DAC$ (Alternate Interior Angles Theorem, since $\overline{BC} \parallel \overline{AD}$ and $\overline{AC}$ is a transversal)
4. $\overline{AC} \cong \overline{AC}$ (Reflexive Property of Congruence)
5. $\triangle ABC \cong \triangle CDA$ (SAS Congruence Postulate, using steps 1, 3, and 4)