Given that the measure of angle 𝐸𝐴𝐷 equals 80 degrees and line segment 𝐴𝐷 is
parallel to line segment 𝐵𝐶, what is the measure of angle 𝐶𝐵𝐸?
We can begin this problem by adding the measure of angle 𝐸𝐴𝐷, which is 80 degrees,
to the diagram and note that we are told that line segments 𝐴𝐷 and 𝐵𝐶 are
parallel. We need to find the measure of angle 𝐶𝐵𝐸. Since we have two parallel lines and the transversal 𝐴𝐵, then the measure of angle
𝐶𝐵𝐸 is simple to calculate. Because these are alternate angles, then the measure of angle 𝐶𝐵𝐸 is equal to the
measure of angle 𝐸𝐴𝐷, and they will both be 80 degrees.
Although we have found the answer, it’s worth taking a closer look at what’s going on
in this particular diagram. And it involves similar triangles. We can recall that similar triangles have corresponding angles congruent and
corresponding sides in proportion. But it might be easy to think that we don’t have enough information to prove that
these two triangles are similar. However, let’s consider the angles. If we can prove that all the corresponding pairs of angles in these triangles are
congruent, then the triangles are similar. And we can prove this even without knowing that angle 𝐸𝐴𝐷 is 80 degrees.
We do already know that because of the parallel lines and the transversal, angles
𝐶𝐵𝐸 and 𝐷𝐴𝐸 will be congruent because these are alternate angles. Angles 𝐵𝐶𝐸 and 𝐴𝐷𝐸 are also alternate angles, and so these are congruent. And finally, angles 𝐶𝐸𝐵 and 𝐷𝐸𝐴 are congruent because they are opposite
angles. This is sufficient to prove that triangles 𝐶𝐵𝐸 and 𝐷𝐴𝐸 are similar. And therefore, if we want to find the measure of angle 𝐶𝐵𝐸, we know that it will
be the same as angle 𝐷𝐴𝐸. And we were given that angle 𝐸𝐴𝐷 is 80 degrees.
Notice that it doesn’t matter if we give this angle as angle 𝐷𝐴𝐸 or 𝐸𝐴𝐷, so
long as the letter 𝐴 is in the middle to represent that the vertex of the angle is
at point 𝐴. And because the corresponding angle 𝐶𝐵𝐸 is congruent, it also has a measure of 80
degrees, which confirms the original answer.