Chapter 3, Problem 7CR

To determine

To prove : The congruency of $\angle A$ with respect to $\angle DBC$ .

Explanation of Solution

Given information : Here, we have given that $\angle A$ is complementary to $\angle BCA$ , $\angle D$ is complementary to $\angle DBC$ and $\angle D\cong \angle BCA$

Proof: As, $\angle A$ is complementary to $\angle BCA$ ,

  $\Rightarrow \angle A+\angle BCA\cong 90°$

  $\Rightarrow \angle BCA\cong 90°-\angle A$

As, $\angle D\cong \angle BCA$ ,

  $\Rightarrow \angle D\cong 90°-\angle A$

Similarly, $\angle D$ is complementary to $\angle DBC$ ,

  $\begin{array}{l}\Rightarrow \angle D+\angle DBC\cong 90°\\ \Rightarrow \angle D\cong 90°-\angle DBC\end{array}$

Now, as, $\angle D\cong 90°-\angle A$ and $\angle D\cong 90°-\angle DBC$ ,

  $\begin{array}{l}\Rightarrow 90°-\angle A\cong 90°-\angle DBC\\ \Rightarrow \angle A\cong \angle DBC\end{array}$