Question

in the figure, \\( \overrightarrow{ba} \\) and \\( \overrightarrow{bc} \\) are opposite rays. \\( \overrightarrow{bf} \\) bisects \\( \angle ebc \\) and \\( \overrightarrow{bh} \\) bisects \\( \angle abf \\).
if \\( m\angle abf=(8 x - 6)^{circ} \\) and \\( m\angle abe=2(x + 11)^{circ} \\), find \\( m\angle ebf \\).

Explanation:

Step1: Use the angle - bisector property

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

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 \(m\angle EBF\)

Since \(m\angle EBF=m\angle ABE\) and \(m\angle ABE = 2(w + 11)\).
Substitute \(w=\frac{25}{2}\) into \(m\angle ABE\):
\(m\angle ABE=2(\frac{25}{2}+11)\).
First, simplify 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:

\(47^{\circ}\)