What is the measure of angle 𝑍𝐹𝑌?
The first thing we want to do is identify angle 𝑍𝐹𝑌. Angles can be named by three points. This one is named 𝑍𝐹𝑌 and that means this is the angle we’re interested in. We want to measure this space. To do that, we’ll have to remember some rules about intersecting lines.
One of the rules we’ll need to remember is that when two lines intersect, opposite angles have the same measure. And we call those opposite congruent angles “verticals angles.” Here’s an example: angle 𝐴𝐶𝐵 is an opposite angle to angle 𝐹𝐶𝑋. So angle 𝐹𝐶𝑋 is equal. It has the same measure as angle 𝐴𝐶𝐵. These two angles are vertical angles. We can label angle 𝑋𝐶𝐹 as 32 degrees.
Another rule that can help us here is the fact that the sum of the interior angles in a triangle measure 180 degrees. Inside this figure, we have a triangle — 𝐶𝐹𝑋. And if we add up all the angles inside this triangle, they will equal 180 degrees. We have 32 degrees. This symbol represents a right angle. And right angles measure 90 degrees plus our unknown measure of angle 𝑋𝐹𝐶 is equal to 180 degrees.
We can add our first two angles together: 32 plus 90 equals 122 degrees. 122 degrees plus our missing angle has to equal 180 degrees. If we take 180 degrees and subtract the 122 degrees, we come up with 58 degrees. Angle 𝑋𝐹𝐶 has a measure of 58 degrees.
But we are still looking for the measure of angle 𝑍𝐹𝑌, here in pink. To find that, we’ll go back to the first rule we started with. We need to go back to what we know about vertical angles. When two lines intersect, opposite angles have the same measure. Our unknown angle 𝑍𝐹𝑌 is a vertical angle of 𝑋𝐹𝐶. Angle 𝑍𝐹𝑌 must be equal to a measure of 58 degrees.
Angle 𝑍𝐹𝑌 equals 58 degrees.