Question

in the diagram below, $overline{ac}paralleloverline{dg}$, $overline{be}congoverline{bf}$ and $mangle abh = 66^{circ}$. find $mangle abe$.

Explanation:

Step1: Identify vertical - angles

Since vertical angles are equal, $\angle ABH$ and $\angle EBI$ are vertical angles. So $m\angle EBI = 66^{\circ}$.

Step2: Use the property of congruent segments

Given $BE\cong BF$, triangle $BEF$ is isosceles. Let the line $BI$ be the angle - bisector of $\angle EBF$ (a line that divides an isosceles triangle into two congruent right - triangles and bisects the vertex angle). Then $\angle ABE=\frac{1}{2}m\angle EBI$.

Step3: Calculate $\angle ABE$

$m\angle ABE=\frac{1}{2}\times66^{\circ}=33^{\circ}$.

Answer:

$33^{\circ}$