Question

in the figure, \\( \overrightarrow { b a } \\) and \\( \overrightarrow { b c } \\) are opposite rays. \\( \overrightarrow { b h } \\) bisects \\( \angle e b c \\) and \\( \overrightarrow { b e } \\) bisects \\( \angle a b f \\).

if \\( m \angle a b e = ( 2 n + 7 ) ^ { \circ } \\) and \\( m \angle e b f = ( 4 n - 13 ) ^ { \circ } \\), find \\( m \angle a b e \\).

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:

\(27^{\circ}\)