Question

given: \\( \overline { b c } \cong \overline { d c } ; \overline { a c } \cong \overline { e c } \\) prove: \\( \triangle b c a \cong \triangle d c e \\)

Explanation:

Step1: Identify vertical angles

$\angle BCA$ and $\angle DCE$ are vertical angles. Vertical angles are equal, so $\angle BCA=\angle DCE$.

Step2: Use SAS (Side - Angle - Side) congruence criterion

We are given that $BC = DC$ and $AC=EC$. Also, from Step1, $\angle BCA=\angle DCE$.
By the SAS (Side - Angle - Side) congruence criterion, if two sides and the included angle of one triangle are equal to two sides and the included angle of another triangle, then the two triangles are congruent.
For $\triangle BCA$ and $\triangle DCE$, we have:

• Side: $BC = DC$
• Angle: $\angle BCA=\angle DCE$
• Side: $AC = EC$

Answer:

$\triangle BCA\cong\triangle DCE$ (by SAS congruence criterion)