Question

in the figure (not drawn to scale), \\( \overline { m o } \\) bisects \\( \angle l m n \\). \\( m \angle l m o = 16 x - 45 \\), and \\( m \angle n m o = x + 105 \\). solve for \\( x \\) and find \\( m \angle l m n \\). a 4.19 b 10.230 c 4.55 d 10.205

Explanation:

Step1: Use the angle - bisector property

Since \(\overrightarrow{MO}\) bisects \(\angle LMN\), then \(m\angle LMO=m\angle NMO\).
So, \(16x - 45=x + 105\).

Step2: Solve the equation for \(x\)

Subtract \(x\) from both sides: \(16x-x-45=x - x+ 105\), which gives \(15x-45 = 105\).
Add \(45\) to both sides: \(15x-45 + 45=105 + 45\), so \(15x=150\).
Divide both sides by \(15\): \(x=\frac{150}{15}=10\).

Step3: Find \(m\angle LMN\)

\(m\angle LMN=m\angle LMO + m\angle NMO\).
Substitute \(x = 10\) into \(m\angle LMO=16x-45\) and \(m\angle NMO=x + 105\).
\(m\angle LMO=16\times10-45=160 - 45 = 115\), \(m\angle NMO=10 + 105=115\).
\(m\angle LMN=115+115 = 230\).

Answer:

B. \(10,230\)