Chapter 3.5, Problem 9PSB

To determine

To prove:Two given line segments are congruent

Explanation of Solution

Given information In a given figure

One given line segment bisects the other given line segment, i.e., $\overline{\text{YW}}\text{ bisects }\overline{\text{AX}}$

One pair of angles is congruent to other, i.e., $\angle \text{A}\cong \angle \text{X}$

Other pair of angles is congruent to other, i.e., $\angle 5\cong \angle 6$

Formula used Bisector divides the line segment in two equal parts

Congruency means same size and same shape

ASA criteria of congruency

Congruent parts of congruent triangle are congruent

Proof

  $\begin{array}{l}\text{In }\Delta \text{AWY and }\Delta \text{WZX, we have}\\ \angle \text{A}\cong \angle \text{X}\mathrm{.....}\text{given}\\ \angle \text{A=}\angle \text{X}\mathrm{....}\text{by definition of congruency}\left(i\right)\\ \angle 5\text{ = }\angle \mathrm{6......}\text{given}\\ \angle 5\text{ = }\angle \mathrm{6.......}\text{by definition of congruency}\mathrm{........}\left(ii\right)\\ \mathrm{Sin}ce,\\ \overline{\text{YW}}\text{ bisects }\overline{\text{AX}}\\ \therefore \overline{\text{AW}}\text{ = }\overline{\text{WX}}\mathrm{......................}\left(iii\right)\\ From\left(i\right),\left(ii\right)\text{and}\left(iii\right),\text{we have}\\ \Delta \text{AWY }\cong \Delta \text{WZX}\mathrm{.......}\text{ASA criteria is satisfied}\\ \overline{\text{ZW}}\cong \overline{\text{YW}}\mathrm{......}\text{Congruent parts of congruent triangle are congruent}\\ \text{Hence proved}\end{array}$