Chapter 3.5, Problem 4PSA

To determine

To prove:Two pairs of angles are congruent in a given triangle.

Explanation of Solution

Given information In a given triangle ACD,

  $\begin{array}{l}\angle 1\cong \angle 3\\ \angle 2\cong \angle 4\end{array}$

Formula used Definition of congruency states objects which have same size and same shape are congruent

Angle − Side − Angle criteria of congruency

Congruent Part of Congruent Triangles are Congruent

Proof

  $\begin{array}{l}\text{In }\Delta \text{BCD and }\Delta \text{ECD},\text{we have}\\ \angle 2\cong \angle \mathrm{4......}\text{Given}\\ \therefore \angle 2=\angle \mathrm{4......}\text{by de`finition of congruency}\mathrm{.....}\left(i\right)\\ \angle 1\cong \angle \mathrm{3........}\text{Given}\\ \therefore \angle 1+\angle 2=\angle 3+\angle 4\\ \angle \text{BCD =}\angle \text{ECD}\mathrm{.......}\left(ii\right)\\ \overline{\text{CD}}\text{ =}\overline{\text{ CD}}\mathrm{......}\text{common side of both the triangles}\mathrm{.....}\left(iii\right)\\ \text{From }\left(i\right),\left(ii\right)\text{ and }\left(iii\right),\text{we have}\\ \Delta \text{BCD }\cong \Delta \text{ECD}\mathrm{......}\text{ASA criteria of congruency satisfied}\\ \overline{\text{BC}}\cong \overline{\text{ED}}\mathrm{........}\text{CPCTC}\\ \text{Hence proved}\end{array}$