Question

in the figure below, ( mangle ebd = 28^circ ), ( overline{bc} ) bisects ( angle abd ), and ( overline{be} ) bisects ( angle cbd ). find ( mangle abc ).

Explanation:

Step1: Analyze BE's role

BE bisects \( \angle CBD \), so \( m\angle CBD = 2 \times m\angle EBD \). Given \( m\angle EBD = 28^\circ \), then \( m\angle CBD = 2 \times 28^\circ = 56^\circ \).

Step2: Analyze BC's role

BC bisects \( \angle ABD \), so \( m\angle ABC = m\angle CBD \) (since BC splits \( \angle ABD \) into two equal angles, and \( \angle ABC \) and \( \angle CBD \) are those equal parts). Thus, \( m\angle ABC = 56^\circ \).

Answer:

\( 56^\circ \)