Chapter 1, Problem 38RP

To determine

To prove: The concurrence of $\angle ABD\approx \angle CBE$ has to be proved

Explanation of Solution

Given information: The following information has been used

  $\angle ABC$ is right angle

  $\angle DBE$ is right angle

Formula used: If BD cuts $\angle ABC$ in between, then

  $\angle ABD+\angle DBC=\angle ABC$

Proof: Now, we use the same formula that we mentioned here to prove the following.

BD cuts $\angle ABC$ in between, so

  $\angle ABD+\angle DBC=\angle ABC$

BC cuts $\angle DBE$ in between, so

  $\angle DBC+\angle CBE=\angle DBE$

Now, as we are given that $\angle ABC$ is right angle and $\angle DBE$ is also right angle, we equate the two of them

  $\angle ABC\approx \angle DBE$

We substitute the values of $\angle ABC$ and $\angle DBE$ from eq. 1 and 2 :

  $\angle ABD+\angle DBC\approx \angle DBC+\angle CBE$

Now, as $\angle DBC$ is common on both sides, we subtract it from each side.

Thus we get :

  $\angle ABD\approx \angle CBE$