Chapter 3, Problem 20CR

To determine

To calculate: The congruency of $\overline{AB}$ with respect to $\overline{CD}$

Explanation of Solution

Given information: $\angle AEC\cong \angle \text{BED}$ , $\overline{AE}\cong \overline{ED}$

Formula used: ASA Congruency rule.

Proof: In $\Delta AEB$ and $\Delta CED$

  $\overline{AE}\cong \overline{ED}$ (Given)

  $\angle BAE\cong \angle CDE$ (Angle opposite to equal lines are equal)

Now, $\angle AEC\cong \angle \text{BED}$ (Given)

  $\angle AEC-\angle BEC\cong \angle \text{BED}-\angle BEC$ (Subtracting $\angle BEC$ from both sides)

  $\Rightarrow \angle AEB\cong \angle CED$

  $\therefore \Delta AEB\cong \Delta CED$ (By angle side angle congruency rule)

  $\therefore \overline{AB}\cong \overline{CD}$ (By Corresponding parts of congruent triangle)

  $$