Question

if ( mangle lmn = 164^circ ), ( mangle fmn = (18x - 6)^circ ), and ( mangle lmf = (23x + 6)^circ ), find ( mangle lmf ).

Explanation:

Step1: Set up angle addition equation

From the diagram, \( \angle LMN = \angle LMF + \angle FMN \). So we substitute the given angle measures:
\( 164 = (23x + 6) + (18x - 6) \)

Step2: Simplify and solve for \( x \)

Combine like terms:
\( 164 = 23x + 18x + 6 - 6 \)
\( 164 = 41x \)
Divide both sides by 41:
\( x = \frac{164}{41} = 4 \)

Step3: Find \( m\angle LMF \)

Substitute \( x = 4 \) into \( m\angle LMF = (23x + 6)^\circ \):
\( m\angle LMF = 23(4) + 6 = 92 + 6 = 98 \)

Answer:

\( 98^\circ \)