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 f = ( 8 w - 6 ) ^ { \circ } \\) and \\( m \angle a b e = 2 ( w + 11 ) ^ { \circ } \\), find \\( m \angle e b f \\).

Explanation:

Step1: Use angle - bisector property

Since \(\overrightarrow{BE}\) bisects \(\angle ABF\), we have \(m\angle ABE=m\angle EBF\) and \(m\angle ABF = 2m\angle ABE\).
Given \(m\angle ABF=(8w - 6)^{\circ}\) and \(m\angle ABE=[2(w + 11)]^{\circ}\), then \((8w-6)=2\times[2(w + 11)]\).

Step2: Solve the equation for \(w\)

Expand the right - hand side: \(8w-6 = 4(w + 11)\).
Using the distributive property \(a(b + c)=ab+ac\), we get \(8w-6=4w+44\).
Subtract \(4w\) from both sides: \(8w-4w-6=4w-4w + 44\), which simplifies to \(4w-6=44\).
Add \(6\) to both sides: \(4w-6 + 6=44+6\), so \(4w=50\).
Divide both sides by \(4\): \(w=\frac{50}{4}=\frac{25}{2}\).

Step3: Calculate \(m\angle ABE\)

Substitute \(w = \frac{25}{2}\) into \(m\angle ABE=[2(w + 11)]^{\circ}\).
\(m\angle ABE=2(\frac{25}{2}+11)\).
First, calculate inside the parentheses: \(\frac{25}{2}+11=\frac{25 + 22}{2}=\frac{47}{2}\).
Then \(m\angle ABE=2\times\frac{47}{2}=47^{\circ}\).

Answer:

\(47^{\circ}\)