In the figure, what is the measure of angle 𝐷𝐹𝑁?
Angle 𝐷𝐹𝑁 is located here. There are multiple ways to solve for the measure of angle 𝐷𝐹𝑁. But they all rely on one thing — the fact that these two lines run parallel. So these here are our parallel lines and a line going through this pair of parallel lines is called the transversal. And there’re properties that we know when using parallel lines and a transversal that will help us solve for the measure of angle 𝐷𝐹𝑁.
So we’re given the angle 𝐵𝑀𝐸, the measure of that is 101 degrees. These two blue angles will be considered alternate exterior angles. They’re on alternate sides of the transversal: one is on the left-hand side and one is on the right. And they’re both on the exterior of the parallel lines. So they’re both outside of the pink lines. And the great thing about alternate exterior angles is they’re congruent. So the measure of angle 𝑁𝐹𝐶 would also be 101 degrees.
Now, these two angles create a line. So we can call them a linear pair or they’re supplementary angles. They add to 180 degrees and the reason why is a straight line is 180 degrees. And measure, so we can call the measure of angle 𝐷𝐹𝑁 — the one that we’re looking for — 𝑥. The measure of angle 𝑁𝐹𝐶 is equal to 101 degrees and we set it equal to 180 degrees. So to solve for 𝑥, we need to subtract 101 degrees from both sides of the equation. And we find that 𝑥 is equal to 79 degrees. Therefore, the measure of angle 𝐷𝐹𝑁 is equal to 79 degrees.
Let’s also show one more way that we could have solved this problem. Instead of using the alternate exterior angles, these two angles in green would be considered corresponding angles. They need to be on the same side of the transversal and they’re both on the left. One needs to be on the inside of the parallel lines and one needs to be on the outside of the parallel lines. And we have that. And the great thing about corresponding angles is they’re also congruent. So the measure of angle of 𝐷𝐹𝑀 would be 101 degrees.
So now, we can still use the fact that the green and the yellow are considered a linear pair because they also make a straight line. So like we said, the measure of angle 𝐷𝐹𝑀 is 101 degrees. And we can let the measure of angle 𝐷𝐹𝑁 what we’re solving for be 𝑥 and set it equal to 180 degrees. So just as before, we solve for 𝑥 by subtracting 101 from both sides of the equation. And once again, we get 79 degrees.
So in this figure, the measure of angle 𝐷𝐹𝑁 is 79 degrees.