Question

in the figure, \\(\overrightarrow{lf}\\) and \\(\overrightarrow{lk}\\) are opposite rays. \\(\overrightarrow{lg}\\) bisects \\(\angle flh\\). if \\(m\angle flg = (14x + 5)\degree\\) and \\(m\angle hlg = (17x - 1)\degree\\), find \\(m\angle flh\\).

Explanation:

Step1: Analyze Angle Bisector

Since \(\overrightarrow{LG}\) bisects \(\angle FLH\), we have \(m\angle FLG = m\angle HLG\).
Given \(m\angle FLG=(14x + 5)^\circ\) and \(m\angle HLG=(17x - 1)^\circ\), set them equal:
\(14x + 5 = 17x - 1\)

Step2: Solve for \(x\)

Subtract \(14x\) from both sides:
\(5 = 3x - 1\)
Add 1 to both sides:
\(6 = 3x\)
Divide by 3:
\(x = 2\)

Step3: Find \(m\angle FLG\)

Substitute \(x = 2\) into \(m\angle FLG=(14x + 5)^\circ\):
\(14(2)+5 = 28 + 5 = 33^\circ\)

Step4: Find \(m\angle FLH\)

Since \(\overrightarrow{LG}\) bisects \(\angle FLH\), \(m\angle FLH = 2\times m\angle FLG\):
\(2\times33^\circ = 66^\circ\)

Answer:

\(66^\circ\)