Given that the measure of angle
𝐹𝐶𝐴 equals 134 degrees in the figure below, two sides are equal. Which are these?
We can begin this problem by adding
the extra angle information to the diagram, namely, that we have a second angle
measure of 134 degrees, which is the measure of angle 𝐹𝐶𝐴. We can also observe that in this
figure we have three parallel line segments. They are 𝐷𝐸, 𝐶𝐹, and 𝐴𝐵. We should be able to identify some
other angle measures by using the properties of parallel lines and transversals. Firstly, we can determine that
angles 𝐵𝐷𝐸 and 𝐷𝐵𝐴 are supplementary, as they are contained within the two
parallel lines and the transversal 𝐵𝐷. These two angles must add to give
180 degrees. Therefore, to find the measure of
angle 𝐷𝐵𝐴, we subtract 134 degrees from 180 degrees, which gives us an angle
measure of 46 degrees.
Then, we can identify another pair
of supplementary angles. This time, considering the parallel
lines 𝐶𝐹 and 𝐴𝐵, we note that angles 𝐹𝐶𝐴 and 𝐶𝐴𝐵 are also
supplementary. So, to find the measure of angle
𝐶𝐴𝐵, we subtract 134 degrees from 180 degrees. And we already know that this will
give us a value of 46 degrees.
We can now consider the question
regarding which two sides are equal. Well, we might not immediately
notice anything about the sides. However, we have worked out that
there are two congruent angles. And if we consider these angles as
part of the triangle 𝐴𝐵𝐶, then these two congruent angles tell us something about
the type of triangle that it is.
By the converse of the isosceles
triangle theorem, we know that if two angles in a triangle are congruent, then the
sides opposite those angles are congruent. And by definition, we have an
isosceles triangle. So, the two congruent sides are
𝐴𝐶 and 𝐵𝐶. Therefore, we can give the answer
that it is side length 𝐴𝐶 and side length 𝐵𝐶 that are equal.