Chapter 3.5, Problem 6PSB

To determine

To prove:That the two-line segments are congruent

Explanation of Solution

Given information In a 4-sided figure

  $\begin{array}{l}\angle \text{1 is complementary to }\angle \text{2}\\ \angle 3\text{ is complementary to }\angle 4\\ \angle 1\cong \angle 3\end{array}$

Formula used Sum of complementary angles is equal to $90°$

SAS criteria of congruency

CPCTC

Proof

  $\begin{array}{l}\text{In }\Delta \text{ABC and }\Delta \text{BCD}\\ \angle \text{1+}\angle \text{2 = }\angle 3\text{ +}\angle 4=90°\\ \left(\text{given that, }\angle \text{1\&}\angle \text{2 and }\angle 3\text{ \&}\angle 4\text{ are complementary}\right)\\ \therefore \angle \text{ABC = }\angle \text{BCD}\mathrm{......}\left(i\right)\\ \text{BC}=\text{BC}\mathrm{.......}\text{common side }\mathrm{.....}\left(ii\right)\\ \text{AC = BD}\mathrm{.....}\text{ sides opposite to equal angles}\mathrm{.....}\left(iii\right)\\ \Delta \text{ABC }\cong \Delta \text{BCD, SAS criteria of congruency is satisfied}\\ \text{⇒}\text{AB}\cong \text{CD, }\mathrm{....}\text{Congruent Parts of Congruent Triangles are Congruent}\\ \text{Hence proved}\end{array}$