Question

given: \\( \overline{ya} \cong \overline{ba} \\), \\( \angle b \cong \angle y \\) prove: \\( \overline{az} \cong \overline{ac} \\)

Explanation:

Step1: Identify vertical angles

$\angle BAC$ and $\angle YAZ$ are vertical angles. By the vertical - angles theorem, $\angle BAC\cong\angle YAZ$.

Step2: Apply the SAS (Side - Angle - Side) congruence criterion

In $\triangle BAC$ and $\triangle YAZ$:

• We know that $BA\cong YA$ (given).
• $\angle BAC\cong\angle YAZ$ (from Step 1).
• $AC\cong AZ$ (given).

Since two sides and the included angle of $\triangle BAC$ are congruent to two sides and the included angle of $\triangle YAZ$, by the SAS congruence criterion, $\triangle BAC\cong\triangle YAZ$.

Step3: Conclude the congruence of the desired sides

Because $\triangle BAC\cong\triangle YAZ$, corresponding parts of congruent triangles are congruent. So, $BC\cong YZ$.

Answer:

By using the vertical - angles theorem and the SAS congruence criterion, we have proved that $BC\cong YZ$.