Question

$B$ lies on $\overline{AC},\overline{BD}$ bisects $\angle ABE$, and $\overline{BE}$ bisects $\angle CBD$ as shown in the above figure. Find the measure of $\angle DBE$.

• A. ${90}^{\circ }$
• B. ${60}^{\circ }$
• C. ${45}^{\circ }$
• D. ${30}^{\circ }$
• E. Cannot be determined from the given information

Answer 1

Answer: (B)

$\angle DBE=\frac{\angle ABE}{2}=\frac{\angle CBE}{2}$

Hence $\angle CBE=ABE$

$\Rightarrow \angle ABD+\angle DBE=\angle CBE+\angle DBE$

$\Rightarrow \angle ABD=\angle CBE$
$\Rightarrow \angle ABD+\angle DBE+\angle CBE={180}^0$

$\Rightarrow 2\angle ABD+\angle DBE={180}^0$

$\Rightarrow \angle ABE+\angle DBE={180}^0$

$\Rightarrow 2\angle DBE+\angle DBE={180}^0$

$\Rightarrow 3\angle DBE={180}^0$

$\Rightarrow \angle DBE={60}^0$