Chapter 2, Problem 38RP

To determine

To prove: Angle CBE is complementary to angle BDC

Explanation of Solution

Given information:

  $\begin{array}{l}\angle ABDissupplementaryto\angle EDC\\ \angle ABDisbisectedby\overrightarrow{BC}\\ \angle BDEisbisectedby\overrightarrow{DC}\end{array}$

Proof:

  $\begin{array}{l}Let\angle ABD=2\theta ,then\angle CBD=\theta (\because BCbisects\angle ABD)\\ \angle BDE=180-2\theta (\because \angle ABDand\angle EDBaresupplementary)\\ \angle BDC=\frac{1}{2}\left(\angle BDE\right)\\ \angle BDC=\frac{1}{2}\left(180-2\theta \right)=90-\theta \\ \therefore \angle CBD+\angle BDC=\theta +90-\theta =90\end{array}$

Hence, $\angle CBDand\angle BDC$ are complementary.