In the figure below, given that the
measure of angle 𝐴𝐶𝐸 equals 131 degrees, find the measure of angle 𝐶𝐸𝐹.
Let’s start by marking on the angle
measure of 𝐴𝐶𝐸, which is 131 degrees, onto the diagram. The measure of angle 𝐶𝐸𝐹 that we
need to calculate is at the bottom of the diagram. Now, we are given that there are
three line segments, 𝐴𝐵, 𝐷𝐶, and 𝐸𝐹, which are all marked as parallel. And that will allow us to work out
some other angle measures.
If we take the top two parallel
lines and the transversal of line segment 𝐴𝐶, then the two angles colored green —
that’s angle 𝐶𝐴𝐵 and angle 𝐴𝐶𝐷 — are alternate interior angles. And we can recall that if a
transversal cuts a pair of parallel lines, then the alternate interior angles are
equal in measure. So, angles 𝐴𝐶𝐷 and 𝐶𝐴𝐵 are
equal in measure. And as the measure of angle 𝐶𝐴𝐵
is 95 degrees, then the measure of angle 𝐴𝐶𝐷 is also 95 degrees.
We can then work out the measure of
the remaining part of this angle 𝐴𝐶𝐸, the angle 𝐷𝐶𝐸. Its measure will be found by
subtracting 95 degrees from 131 degrees, which is 36 degrees. So, to find the measure of angle
𝐶𝐸𝐹, we can use the same property of parallel lines again. This time, we can consider the
parallel line segments 𝐷𝐶 and 𝐸𝐹 and the transversal of line segment 𝐶𝐸. Then, we can recognize that the
angles 𝐷𝐶𝐸 and 𝐶𝐸𝐹 are alternate interior angles. And once again, we know that these
angles must be equal in measure. So they are both 36 degrees.
Therefore, by using the properties
of parallel lines and transversals, we have determined that the measure of angle
𝐶𝐸𝐹 is 36 degrees.