Question

in the figure shown, suppose ( mangle abc=n ) and ( mangle abd = 2(mangle dbc) ). the angle bisector of ( angle dbc ) is ( overrightarrow{be} ). what is ( mangle ebc )?

( mangle ebc=square )
(do not include the degree symbol in your answer.)

Explanation:

Step1: Find \(m\angle DBC\)

Since \(m\angle ABC = m\angle ABD + m\angle DBC\) and \(m\angle ABD = 2m\angle DBC\), and \(m\angle ABC=n\).
Substitute \(m\angle ABD\) into the equation: \(n=2m\angle DBC + m\angle DBC\).
So, \(n = 3m\angle DBC\), then \(m\angle DBC=\frac{n}{3}\).

Step2: Use the angle - bisector property

Because \(\overrightarrow{BE}\) is the angle bisector of \(\angle DBC\), by the definition of an angle bisector \(m\angle EBC=\frac{1}{2}m\angle DBC\).
Substitute \(m\angle DBC=\frac{n}{3}\) into the formula: \(m\angle EBC=\frac{1}{2}\times\frac{n}{3}\).

Answer:

\(\frac{n}{6}\)