Question

in the figure below, ( mangle abd = 101^{circ}, mangle cbd = 68^{circ} ), and ( overrightarrow{be} ) bisects ( angle cbd ). find ( mangle abe ).

Explanation:

Step1: Find the measure of \(\angle CBE\)

Since \(BE\) bisects \(\angle CBD\), by the angle - bisector definition, \(\angle CBE=\frac{1}{2}\angle CBD\).
Given \(\angle CBD = 68^{\circ}\), then \(\angle CBE=\frac{68^{\circ}}{2}=34^{\circ}\).

Step2: Find the measure of \(\angle ABE\)

By the angle - addition postulate, \(\angle ABE=\angle ABD-\angle EBD\).
We know that \(\angle ABD = 101^{\circ}\) and \(\angle EBD=\angle CBE = 34^{\circ}\).
So \(\angle ABE=101^{\circ}-34^{\circ}\).

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Answer:

\(67\)