Chapter 3, Problem 23CR

To determine

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

Explanation of Solution

Given information: $\odot O,\overline{OD}\cong \overline{OE},\angle DOB\cong \angle EOA$ , $\overline{AE}\cong \overline{ED}$

Formula used: SAS congruency rule.

Proof: In $\Delta AOD$ and $\Delta EOB$

  $\overline{OD}\cong \overline{OE}$ (Given)

  $\overline{AO}\cong \overline{BO}$ (Radius of same circle are equal)

Now, $\angle DOB\cong \angle EOA$ (Given)

  $\angle DOB-\angle EOD\cong \angle EOA-\angle EOD$ (Subtracting $\angle EOD$ from both sides)

  $\therefore \angle EOB\cong \angle DOA$

  $\therefore \Delta AOD\cong \Delta EOB$ (By side angle side congruency rule)

  $\angle DAO\cong \angle EBO$ (Corresponding parts of congruent triangle are equal)

Or $\angle CAO\cong \angle CBO$

  $\therefore \overline{CA}\cong \overline{CB}$ (Sides opposite to equal angles are equal)

  $$