Question

19 ray mo bisects ∠lmn, ∠lmo = 8x - 23, and ∠nmo = 2x + 31. solve for x.

1. ray mo bisects ∠lmn, ∠lmo = 8x - 23, and ∠nmo = 2x + 31. find the measure of ∠lmo.

Explanation:

Step1: Use angle - bisector property

Since ray \(MO\) bisects \(\angle LMN\), then \(\angle LMO=\angle NMO\).
So, \(8x - 23=2x + 31\).

Step2: Solve the equation for \(x\)

Subtract \(2x\) from both sides: \(8x-2x - 23=2x-2x + 31\), which gives \(6x-23 = 31\).
Add \(23\) to both sides: \(6x-23 + 23=31 + 23\), so \(6x=54\).
Divide both sides by \(6\): \(x=\frac{54}{6}=9\).

Step3: Find the measure of \(\angle LMO\)

Substitute \(x = 9\) into \(\angle LMO=8x - 23\).
\(\angle LMO=8\times9-23\).
First, calculate \(8\times9 = 72\). Then \(72-23 = 49\).

Answer:

For question 19: \(x = 9\) (Option A).
For question 20: \(\angle LMO = 49^{\circ}\) (Option C).