Final answer:
The value of angle MNQ, which is bisected by line NF, is 78 degrees. This is found by setting 8x + 12 equal to two times the measure of FNQ, which is 39 degrees, and solving for x.
Step-by-step explanation:
To find the value of angle MNQ, we need to use the information that line NF bisects angle MNQ meaning MNF and FNQ are equal, m(MNQ) = 8x + 12, and that angle PNQ = 78 degrees. We must also incorporate that angle RNM = 3y - 9, though it's not directly part of finding the measure of MNQ.
Since NF bisects MNQ, we can set up the equation:
8x + 12 = 2 * m(FNQ).
Because PNQ is part of MNQ and is 78 degrees, FNQ is half of PNQ so m(FNQ) = 78/2 = 39 degrees.
Next, substitute the value into the equation and solve for x:
8x + 12 = 2 * 39
8x + 12 = 78
8x = 66
x = 8.25.
Now, use the value of x to find m(MNQ):
m(MNQ) = 8(8.25) + 12 = 66 + 12 = 78 degrees.