Video Transcript
Given that 𝐴𝐵𝐶𝐷 is a square, find the measure of angle 𝑌𝐶𝐵.
Let’s begin by considering the diagram, including identifying where the angle 𝑌𝐶𝐵
is that we are asked to find out. We can observe from the markings on the diagram that there are two congruent line
segments, 𝐵𝑌 and 𝐶𝑋. Given that we have some information about congruent line segments, we can consider
the two triangles 𝑌𝐵𝐶 and 𝑋𝐶𝐷. In particular, we might pose the question, is triangle 𝑋𝐶𝐷 congruent to triangle
𝑌𝐵𝐶?
It can sometimes be difficult to visually separate things on a diagram. So a separate sketch of the triangles can be a useful tool in problems like this. We just need to make sure we include all the relevant information about lengths and
angles. So, given that 𝐴𝐵𝐶𝐷 is a square, that means we can say that all the angles at its
vertices are 90 degrees. And importantly, within the triangles, we know that the angles 𝑌𝐵𝐶 and 𝑋𝐶𝐷 both
have measures of 90 degrees. So these are right triangles. And since 𝐴𝐵𝐶𝐷 is a square, that means that all its sides are the same
length. So the sides 𝐵𝐶 and 𝐶𝐷 in our triangles are congruent.
We now have enough information to apply one of the congruence criteria. We have two pairs of corresponding sides congruent, and the included pair of angles
are congruent. We have therefore proved that triangles 𝑋𝐶𝐷 and 𝑌𝐵𝐶 are congruent by the SAS
congruency criterion.
Notice that although we do have two right triangles, we don’t have the information
about the lengths of the hypotenuse in each triangle. So we couldn’t prove the triangles are congruent by using the RHS criterion.
So let’s return to the question of finding the measure of angle 𝑌𝐶𝐵. As the triangles are congruent, we can say that the corresponding and therefore
congruent angle in triangle 𝑋𝐶𝐷 will be that of angle 𝑋𝐷𝐶. And since we have a square, we know that angle 𝐴𝐷𝐶 is a right angle. So the measure of angle 𝑋𝐷𝐶 is equal to 90 degrees minus 56 degrees, which is 34
degrees. The angle measure of 𝑌𝐶𝐵 is equal to this. So we can give the answer that the measure of angle 𝑌𝐶𝐵 is 34 degrees.
