Question Video: Understanding the Relation between Sides and Corresponding Angles in Triangles Mathematics

Video Transcript

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.