Chapter 3.4, Problem 8PSB

To determine

To prove: The AC and BD line segments are congruent in the given triangle.

Explanation of Solution

Given information:

Two given angles are congruent, i.e., $\angle \text{AEB }\cong \angle \text{DEC}$

Two given line segments are congruent, i.e., $\overline{\text{AE}}\cong \overline{\text{DE}}$

And

Two given angles are congruent, i.e., $\angle \text{A}\cong \angle \text{D}$

Formula used :

Same size and same shape objects are always congruent, as per definition of congruency

ASA criteria of congruency

CPCTC

Proof :

  $\begin{array}{l}\text{In }\Delta \text{ AEC and }\Delta \text{ DEB, }\\ \angle \text{ AEB }\cong \angle \text{ DEC}\mathrm{........}\text{Given}\\ \therefore \angle \text{ AEB = }\angle \text{ DEC}\mathrm{.......}\text{ by definition of congruency}\\ \text{⇒}\angle \text{ AEB + }\angle \text{BEC = }\angle \text{ DEC +}\angle \text{BEC }\\ \text{⇒}\angle \text{ AEC = }\angle \text{ DEB}\mathrm{..........}\left(i\right)\\ \overline{\text{AE}}\cong \overline{\text{DE}}\mathrm{..........}\text{Given}\\ \therefore \overline{\text{AE}}=\overline{\text{DE}}\mathrm{.........}\left(ii\right)\mathrm{....}\text{by definition of congruency}\\ \angle \text{A}\cong \angle \text{D}\mathrm{...........}\text{ Given}\\ \angle \text{A=}\angle \text{D}\mathrm{.........}\text{ by definition of congruency}\mathrm{......}\left(iii\right)\\ \text{From }\left(i\right),\left(ii\right)\text{ and }\left(iii\right),\text{we get}\\ \therefore \Delta \text{ AEC }\cong \Delta \text{ DEB}\mathrm{.......}\text{ASA criteria of congruency is satisfied}\\ \text{⇒}\overline{\text{AC}}\cong \overline{\text{BD}}\mathrm{..........}CPCTC\\ \text{Hence proved}\end{array}$