Question

1. if $overrightarrow{mo}$ bisects $angle pmn$, $mangle pmn = 74^circ$ and $mangle omn = 2x + 7$, find the value of x.

Explanation:

Step1: Recall Angle Bisector Definition

An angle bisector divides an angle into two equal parts. So, \( \overrightarrow{MO} \) bisecting \( \angle PMN \) means \( m\angle OMN=\frac{1}{2}m\angle PMN \).

Step2: Substitute Given Values

We know \( m\angle PMN = 74^\circ \) and \( m\angle OMN=2x + 7 \). Substitute into the equation: \( 2x+7=\frac{1}{2}\times74 \).

Step3: Simplify Right - Hand Side

Calculate \( \frac{1}{2}\times74 = 37 \), so the equation becomes \( 2x + 7=37 \).

Step4: Solve for x

Subtract 7 from both sides: \( 2x=37 - 7=30 \). Then divide both sides by 2: \( x=\frac{30}{2}=15 \).

Answer:

\( x = 15 \)