Final answer:
By setting the expressions for LMO and NMO equal, solving for x, and substituting back, we find that x equals 11 and the angle LMN equals 170 degrees.
Step-by-step explanation:
To solve for x and find the angle LMN, we must set the two expressions for the angles LMO and NMO equal to each other since MO bisects LMN, meaning LMO and NMO are equal.
The equation is 9x - 14 = x + 74. To solve for x, we first move the x terms to one side of the equation: 9x - x = 74 + 14, which simplifies to 8x = 88. Dividing both sides by 8, we get x = 11.
To find the angle LMN, we substitute x back into one of the original expressions. For LMO, 9x - 14 becomes 9(11) - 14, which equals 85 degrees. Since LMO and NMO are equal, LMN is twice the measure of one of these angles, so LMN = 2 * 85 = 170 degrees.