Question

given: \\( \overline { d e } \cong \overline { c e } \\), \\( \overline { a d } \perp \overline { d b } \\) and \\( \overline { b c } \perp \overline { c a } \\).
prove: \\( \triangle a e b \\) is an isosceles triangle.

( \begin{array} { | c | c | c | } hline \text { step } & \text { statement } & \text { reason } \\ hline & overline { d e } \cong overline { c e } & \\ & overline { a d } perp overline { d b } & \text { given } \\ & overline { b c } perp overline { c a } & \\ hline end{array} )

Explanation:

Step1: Prove $\triangle ADE\cong\triangle BCE$

• $\angle ADE = \angle BCE=90^{\circ}$ (Definition of perpendicular lines)
• $\angle AED=\angle BEC$ (Vertical angles are congruent)
• $\overline{DE}\cong\overline{CE}$ (Given)
• By AAS (Angle - Angle - Side) congruence criterion, $\triangle ADE\cong\triangle BCE$

Step2: Find the relationship between $AE$ and $BE$

• Since $\triangle ADE\cong\triangle BCE$, then $\overline{AE}\cong\overline{BE}$ (Corresponding parts of congruent triangles are congruent)

Answer:

Since $\overline{AE}\cong\overline{BE}$, $\triangle AEB$ is an isosceles triangle.