Given that 𝐴𝐵𝐶𝐷 is a square, find, in degrees, the values of 𝑥 and 𝑦.
So we have a diagram of 𝐴𝐵𝐶𝐷, which is a square. We also have one of the diagonals of the square, 𝐵𝐷, drawn in. And a line, 𝐴𝐹, which isn’t a diagonal of the square. 𝑥 and 𝑦 represent angles within this diagram. Now we don’t yet have enough information to be able to calculate 𝑥 or 𝑦 directly. So let’s think about any other angles in the diagram that we could find first.
There’s an angle of a 100 degrees marked in the diagram. And in fact this angle is vertically opposite angle 𝐴𝐸𝐵, as these two angles are formed when a pair of straight lines intersect. A key fact about vertically opposite angles is that they are equal or congruent. And therefore, angle 𝐴𝐸𝐵 is also equal to 100 degrees. Now let’s consider the triangle 𝐴𝐸𝐵, in which angle 𝑥 is one of the three angles and another is this angle we’ve just calculated of 100 degrees.
If we could work out the third angle in this triangle, angle 𝐴𝐵𝐸, then we’d be able to use the fact that the angle sum in a triangle is always 180 degrees to find the value of 𝑥. So let’s look at this angle more closely. This angle is in the corner of the square. And it’s formed where the diagonal meets the vertex. A key fact about the diagonals of a square is that they bisect, or cut in half, a pair of opposite angles, in this case angles 𝐴𝐵𝐶 and 𝐴𝐷𝐶, which means that the two portions of this angle are equal. So I’ve marked them both as green arcs.
All of the interior angles in a square are 90 degrees. And therefore the angle 𝐴𝐵𝐸, which we’ve just shown is half of this, is equal to 45 degrees. So now as we wanted, we know two of the angles in triangle 𝐴𝐵𝐸. The third angle 𝑥 is the one we want to calculate. So we can find the value of 𝑥 by subtracting the other two angles in this triangle from 180 degrees. We have 180 degrees minus 100 degrees minus 45 degrees, 𝑥 is 35 degrees. So now that we found the value of 𝑥, we can think about calculating the value of 𝑦.
In order to do so, let’s think about the triangle 𝐴𝐵𝐹, now marked in pink. This triangle includes the sides 𝐴𝐵 and 𝐵𝐹. 𝐴𝐵 is the side of the square. And 𝐵𝐹 is part of a side. And so these two sides are at right angles with each other. It also includes angle 𝑥, which we’ve just shown to be 35 degrees. Our method for calculating 𝑦 then is going to be to calculate the third angle in this triangle, angle 𝐵𝐹𝐴, and then use the fact that it’s on a straight line with 𝑦 in order to find the value of 𝑦.
This is actually equivalent to applying the following fact about triangles. The exterior angle of a triangle is equal to the sum of the other two interior angles. So 𝑦 is an exterior angle. And in order to find it, we need to sum the other two interior angles. So that’s the right angle and the angle of 35 degrees. This gives the value of 𝑦, 125 degrees. Don’t worry if you couldn’t remember that fact, you could do it using the step-by-step approach I mentioned earlier.
First calculate the other interior angle in the triangle, angle 𝐵𝐹𝐴, by using the fact that the angle sum in a triangle is 180 degrees. And then calculate 𝑦 as it’s on the straight line with this angle. And therefore, the sum of these two angles is also 180 degrees. Our solution to the problem then, 𝑥 is equal to 35 degrees. 𝑦 is equal to 125 degrees. Remember, the key fact that we used early on in the question was that the diagonals of a square bisect a pair of opposite angles.