Question

a30 name: _
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 \\)

Explanation:

Step1: State the right - angle information

Since \(\overline{BD}\perp\overline{AB}\) and \(\overline{BD}\perp\overline{DE}\), \(\angle B=\angle D = 90^{\circ}\) (Definition of perpendicular lines).

Step2: Use vertical angles

\(\angle ACB=\angle ECD\) (Vertical angles are congruent).

Step3: Prove triangle congruence

In \(\triangle ABC\) and \(\triangle EDC\), we have \(\angle B=\angle D\), \(\angle ACB=\angle ECD\), and \(BC = DC\) (Given). By the AAS (Angle - Angle - Side) congruence criterion, \(\triangle ABC\cong\triangle EDC\).

Step4: Conclude the angle congruence

Since \(\triangle ABC\cong\triangle EDC\), \(\angle A\cong\angle E\) (Corresponding parts of congruent triangles are congruent).

Answer:

\(\angle A\cong\angle E\) is proved as shown above.