Question

more triangle congruence proofs
for each problem write either a 2 - column or flowchart proof.

1. given: \\( \overline { b d } \perp \overline { a b } , \overline { b d } \perp \overline { d e } \\),\\( \overline { b c } \cong \overline { d c } \\)prove: \\( \angle a \cong \angle e \\)
2. given: \\( \overline { b c } \cong \overline { d c } , \overline { a c } \cong \overline { e c } \\)prove: \\( \delta a b c \cong \delta e d c \\)
3. given: \\( \overline { y a } \cong \overline { b a } , \angle b \cong \angle y \\)prove: \\( \overline { a z } \cong \overline { a c } \\)

Explanation:

Step1: Identify vertical angles

$\angle ACB$ and $\angle ECD$ are vertical angles. By the vertical angles theorem, $\angle ACB\cong\angle ECD$.

Step2: Use the AAS (Angle - Angle - Side) congruence criterion

We know that $\overline{BC}\cong\overline{DC}$ (given), $\angle ABC = \angle EDC=90^{\circ}$ (since $\overline{BD}\perp\overline{AB}$ and $\overline{BD}\perp\overline{DE}$), and $\angle ACB\cong\angle ECD$ (from Step 1).
In $\triangle ABC$ and $\triangle EDC$, we have two angles and a non - included side congruent. So, by the AAS congruence criterion, $\triangle ABC\cong\triangle EDC$.

Answer:

$\triangle ABC\cong\triangle EDC$ (by AAS congruence criterion: $\angle ABC\cong\angle EDC$, $\angle ACB\cong\angle ECD$, $\overline{BC}\cong\overline{DC}$)