Chapter 3.3, Problem 18PSB

To determine

To prove : The congruency of $\overline{FG}$ with respect to $\overline{HG}$

Explanation of Solution

Given information : In this question, we have given that $\overline{KG}\cong \overline{GJ}$ , $\angle 2\cong \angle 4$ , $\angle 1$ is complementary to $\angle 2$ , $\angle 3$ is complementary to $\angle 4$ and $\angle FGJ\cong \angle HGK$ .

Formula used: ASA Congruency rule

Proof: As, it is given that $\angle 1$ is complementary to $\angle 2$ and $\angle 3$ is complementary to $\angle 4$ .

  $\Rightarrow \angle 1+\angle 2=90°$ and $\angle 3+\angle 4=90°$

  $\Rightarrow \angle 1=90°-\angle 2$ and $\angle 3=90°-\angle 4$

Now, as $\angle 2\cong \angle 4$

  $\Rightarrow \angle 1=90°-\angle 2$ and $\angle 3=90°-\angle 2$

  $\therefore \angle 1\cong \angle 3$

As, $\angle FGJ\cong \angle HGK$ (Given)

Subtracting $\angle KGJ$ from both sides, we get,

  $\Rightarrow \angle KGF\cong \angle JGH$

Now, In $\Delta FGJand\Delta HGK$

  $\angle KGF\cong \angle JGH$ (Proved above)

  $\overline{KG}\cong \overline{GJ}$ (Given)

  $\angle 1\cong \angle 3$ (Proved above)

  $\therefore \Delta FGJ\cong \Delta HGK$ (BY ASA Congruency rule)

  $\Rightarrow \overline{FG}\cong \overline{HG}$