Questions: In the diagram below, AC is parallel to DG, BE is congruent to BF, and the measure of angle ABH is 73 degrees. Find the measure of angle ABE.

Transcript text: In the diagram below, $\overline{A C} \| \overline{D G}, \overline{B E} \cong \overline{B F}$ and $\mathrm{m} \angle A B H=73^{\circ}$. Find $\mathrm{m} \angle A B E$.

Solution

Solution Steps

Step 1: Identify Given Information

We are given that $ AC \parallel DG $, $ BE \cong BF $, and $ m\angle ABH = 73^\circ $. We need to find $ m\angle ABE $.

Step 2: Use Parallel Line Properties

Since $ AC \parallel DG $ and $ AB $ is a transversal, the corresponding angles $ \angle ABH $ and $ \angle BHD $ are equal. Therefore, $ m\angle BHD = 73^\circ $.

Step 3: Use Isosceles Triangle Properties

Since $ BE \cong BF $, triangle $ BEF $ is isosceles with $ \angle BEF = \angle BFE $. Let $ x $ be the measure of these angles. The sum of angles in triangle $ BEF $ is $ 180^\circ $, so: \[ \angle BEF + \angle BFE + \angle EBF = 180^\circ \] \[ x + x + 73^\circ = 180^\circ \] \[ 2x + 73^\circ = 180^\circ \] \[ 2x = 107^\circ \] \[ x = 53.5^\circ \]

Final Answer

\[ m\angle ABE = 53.5^\circ \]

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