Chapter 8, Problem 18RP

To determine

To prove:The product of segments is equal i.e. $\left(\text{RC}\right)\left(\text{ZA}\right)\text{=}\left(\text{ZY}\right)\left(\text{RW}\right)$

Explanation of Solution

Given information:

A parallelogram BRXZ and

Two angles are congruent, i.e., $\angle 3\cong \angle 4$

Formula used:

Congruency of triangles

Ratios of corresponding sides are equivalent

Proof:

In triangles RCW and ZAY, we have

. $\angle 3\cong \angle 4$ ……..Given

  $\angle 3=\angle 4$ ……..Congruent angles are always equal……….(i)

RC=ZY……parts of parallel equal sides……….(ii)

ZA=RW,,,,,,,,parts of parallel sides……..(iii)

  $\therefore$

  $\Delta \text{RCW}\cong \Delta \text{ZAY}$ …….SAS criteria is satisfied

Thus,

Ratio of corresponding sides are equivalent and we get

  $\frac{\text{RC}}{\text{RW}}=\frac{\text{ZY}}{\text{ZA}}$

i.e., $\left(RC\right)\left(ZA\right)\text{=}\left(ZY\right)\left(RW\right)$

Hence Proved