Question

in the figure, ( overrightarrow{ba} ) and ( overrightarrow{bc} ) are opposite rays. ( overrightarrow{bh} ) bisects ( angle ebc ) and ( overrightarrow{be} ) bisects ( angle abf ).
if ( mangle abe=(2n + 7)^{circ} ) and ( mangle ebf=(4n - 13)^{circ} ), find ( mangle abe ).
115

Explanation:

Step1: Use the angle - bisector property

Since \(\overrightarrow{BE}\) bisects \(\angle ABF\), then \(m\angle ABE=m\angle EBF\).
So, \(2n + 7=4n-13\).

Step2: Solve the equation for \(n\)

Subtract \(2n\) from both sides: \(2n+7 - 2n=4n - 13-2n\), which gives \(7 = 2n-13\).
Add \(13\) to both sides: \(7 + 13=2n-13 + 13\), so \(20 = 2n\).
Divide both sides by \(2\): \(n=\frac{20}{2}=10\).

Step3: Find \(m\angle ABE\)

Substitute \(n = 10\) into \(m\angle ABE=(2n + 7)^{\circ}\).
\(m\angle ABE=(2\times10 + 7)^{\circ}=(20 + 7)^{\circ}=27^{\circ}\).

Answer:

\(27^{\circ}\)