Given that 𝐴𝐵𝐶𝐷 is a square and the measure of angle 𝐸𝐷𝐶 equals 62 degrees, find the measure of angle 𝐷𝐸𝐴.
Let’s start this question by reminding ourselves of some of the properties of a square. We know that a square has four equal length sides and four equal angles that are 90 degrees each. So in our diagram, the square 𝐴𝐵𝐶𝐷 is split into three different triangles. We’re told that the measure of angle 𝐸𝐷𝐶 is 62 degrees. And we need to work out the angle marked pink, that is, the measure of angle 𝐷𝐸𝐴. If we look at the triangle 𝐴𝐷𝐸, we could, in theory, find our missing angle if we knew the two other angles in this triangle. So let’s have a look and see if we can calculate either of these two angles.
Let’s start by looking at the angle 𝐴𝐷𝐸. We can see that it’s part of the larger angle 𝐴𝐷𝐶, which is one of the angles on our square. So to find 𝐴𝐷𝐸 then, we could find our right angle 𝐴𝐷𝐶 and subtract 62 degrees, which is the angle 𝐸𝐷𝐶. So then we can say that the measure of angle 𝐴𝐷𝐸 must be 28 degrees. Next, let’s see if we can calculate the measure of angle 𝐷𝐴𝐸. We noticed that the line 𝐶𝐴 is a diagonal of our square. We can recall that the diagonals of a square bisect the angles. The word bisect means to split exactly in two. This means that the 90-degree angles at the vertices of our square will be split into two equal 45-degree angles by the diagonal. This means that the measure of angle 𝐷𝐴𝐸 must be equal to 45 degrees.
So let’s now return to our triangle 𝐴𝐷𝐸 and see if we can find the measure of our third missing angle 𝐷𝐸𝐴. We can use the fact that the angles in a triangle add up to 180 degrees. So this means that the measure of angle 𝐴𝐷𝐸 plus the measure of angle 𝐷𝐴𝐸 plus the measure of angle 𝐷𝐸𝐴 must be equal to 180 degrees. We can fill in the angles that we know to give us 28 degrees plus 45 degrees plus the measure of angle 𝐷𝐸𝐴 equals 180 degrees. So 73 degrees plus the measure of angle 𝐷𝐸𝐴 would be a 180 degrees. So to find the measure of angle 𝐷𝐸𝐴, we would subtract 73 from both sides, giving us a 180 degrees minus 73 degrees, giving us 107 degrees. So our final answer for the measure of angle 𝐷𝐸𝐴 is 107 degrees.