Final answer:
The measure of angle LMN is found by adding the measures of angles LMO and NMO, given by the expressions 8x - 14 and 2x + 34 respectively. They form a linear pair that equals 180 degrees. Simplifying yields the expression 10x + 20 equals 180, which, once solved for x, indicates that option (C) 10x - 20 is the correct answer.
Step-by-step explanation:
To find the measure of angle LMN, we need to recognize that LMO and NMO form a linear pair if LMN is the angle formed by the two lines meeting at point M. Since LMO and NMO are on a straight line, their measures should add up to 180 degrees (straight angle). We are given the measures of angles LMO and NMO in terms of x.
LMO = 8x – 14
NMO = 2x + 34
To find LMN, let's add LMO and NMO and set them equal to 180 degrees:
8x - 14 + 2x + 34 = 180
Combining like terms:
10x + 20 = 180
We solve this equation by subtracting 20 from both sides:
10x = 160
And then dividing both sides by 10 to solve for x:
x = 16
Since the question asks for the measure of angle LMN in terms of x, the correct answer is the one that has a coefficient of 10 in front of x since we established that LMO + NMO = 10x + 20.
Therefore, the measure of angle LMN is 10x - 20, answering it as (C) 10x - 20.