Chapter 3.5, Problem 10PSB

To determine

To prove:Two given line segments are congruent

Explanation of Solution

Given information One given point is midpoint of line segment, i.e., B is midpoint of $\overline{\text{AC}}$

Oher given point is midpoint of line segment, i.e., E is midpoint of $\overline{\text{AD}}$

One pair of given angles is congruent, i.e., $\angle 7\cong \angle 8$

Other pair of angles is congruent, i.e., $\angle \text{ECD}\cong \angle \text{BDC}$

Formula used Congruency means same size and same shape

AAS criteria of congruency

Congruent parts of congruent triangle are congruent

Proof

  $\begin{array}{l}\text{In }\Delta \text{BCD and }\Delta \text{ECD, we have}\\ \angle \text{C}\cong \angle \text{D}\mathrm{.....}\text{given}\\ \angle \text{C=}\angle \text{D}\mathrm{....}\text{by definition of congruency}\left(i\right)\\ \angle 7\text{=}\angle \mathrm{8......}\text{given}\\ \angle 7+\angle \text{BCD = }\angle \text{EDC+}\angle 8=180°\mathrm{...}\text{as per figure}\\ \angle \text{BCD = }\angle \text{EDC}\mathrm{.......}\left(ii\right)\\ \text{CD = CD}\mathrm{.....}\text{ common side}\mathrm{.....}\left(iii\right)\\ From\left(i\right),\left(ii\right)\text{and}\left(iii\right),\text{we have}\\ \Delta \text{BCD }\cong \Delta \text{ECD, AAS criteria of congruency is satisfied}\\ \overline{\text{AC}}\cong \overline{\text{AD}}\mathrm{......}\text{Congruent parts of congruent triangle are congruent}\\ \text{Hence proved}\end{array}$