Question

angle abc is a straight angle. ( mangle dbc = 130^{circ} ) and ( overrightarrow{be} ) bisects ( angle abd ). what is ( mangle eba? )

Explanation:

Step1: Find \(m\angle ABD\)

Since \(\angle ABC\) is a straight angle (\(m\angle ABC = 180^{\circ}\)) and \(m\angle DBC=130^{\circ}\), we use the angle - addition postulate \(m\angle ABC=m\angle ABD + m\angle DBC\). Then \(m\angle ABD=m\angle ABC - m\angle DBC\). Substituting the values, we get \(m\angle ABD = 180^{\circ}-130^{\circ}=50^{\circ}\).

Step2: Use the angle - bisector property

Since \(\overrightarrow{BE}\) bisects \(\angle ABD\), by the definition of an angle bisector (\(m\angle EBA=\frac{1}{2}m\angle ABD\)). Substituting \(m\angle ABD = 50^{\circ}\), we have \(m\angle EBA=\frac{1}{2}\times50^{\circ}=25^{\circ}\).

Answer:

\(25\)