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.
step
statement - reason
1
$ \overline { d e } \cong \overline { c e } $
$ \overline { a d } \perp \overline { d b } $
$ \overline { b c } \perp \overline { c a } $
given

Question

Accepted Answer

# Explanation:
## Step1: Prove $\triangle ADE\cong\triangle BCE$
Since $\overline{AD}\perp\overline{DB}$ and $\overline{BC}\perp\overline{CA}$, $\angle ADE = \angle BCE=90^{\circ}$.
We know that $\angle AED=\angle BEC$ (vertical angles are equal) and $\overline{DE}\cong\overline{CE}$ (given).
By the Angle - Angle - Side (AAS) congruence criterion ($\angle ADE=\angle BCE$, $\angle AED = \angle BEC$, $\overline{DE}=\overline{CE}$), $\triangle ADE\cong\triangle BCE$.
## Step2: Find the relationship between $AE$ and $BE$
Because $\triangle ADE\cong\triangle BCE$, corresponding parts of congruent triangles are equal. So, $\overline{AE}\cong\overline{BE}$.
## Step3: Use the definition of an isosceles triangle
A triangle is isosceles if it has at least two congruent sides. Since $\overline{AE}\cong\overline{BE}$ in $\triangle AEB$, by the definition of an isosceles triangle, $\triangle AEB$ is an isosceles triangle.

# Answer:
$\triangle AEB$ is an isosceles triangle.

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QUESTION IMAGE

Question

Explanation:

Step1: Prove \(\triangle ADE\cong\triangle BCE\)

Step2: Find the relationship between \(AE\) and \(BE\)

Step3: Use the definition of an isosceles triangle

Answer: