In the diagram, 𝐴𝐵𝐶𝐷 is a kite, and 𝐵𝐸𝐹𝐶 and 𝐶𝐺𝐻𝐷 are squares. Prove that 𝑥 is equal to three 𝑦.
So our goal is to prove that 𝑥 is equal to three 𝑦. And we have two 𝑦 and 𝑦 in our diagram, along with our angle 𝑥. So to prove this, let’s first go through the given information and try to use properties that might be useful to prove this.
So first, we know that 𝐴𝐵𝐶𝐷 is a kite. So if we have a kite, we can say that angle 𝐵𝐴𝐷 is equal to angle 𝐵𝐶𝐷 because it’s a kite. In a kite, the angles across from each other are congruent. So like we said, angle 𝐵𝐴𝐷 and angle 𝐵𝐶𝐷 would be equal. And since angle 𝐵𝐴𝐷 is equal to 𝑥, we can also call angle 𝐵𝐶𝐷 𝑥.
Next, we are given two squares: 𝐵𝐸𝐹𝐶 and 𝐶𝐺𝐻𝐷. Now instead of using every property of a square and every property of a kite to label this diagram, we’re trying to prove that 𝑥 is equal to three 𝑦 and all of the 𝑥s and 𝑦s that are found in our diagram are angles. So let’s go through the angle information that we could include on our diagram.
So since we have these two squares, in a square each of the angles are equal to 90 degrees. So as we said, each of these angles would be equal to 90 degrees. So they’re all equal to each other and each of them will be 90 degrees because 𝐵𝐸𝐹𝐶 is a square. And because 𝐶𝐺𝐻𝐷 is a square, each of its angles are equal to 90 degrees.
So the only shape that we haven’t touched is triangle 𝐶𝐹𝐺. But we were given that angle 𝐶𝐹𝐺 was equal to two 𝑦 and angle 𝐹𝐺𝐶 is equal to 𝑦. And all angles in a triangle sum to 180 degrees. So this missing angle 𝐹𝐶𝐺, we could say, we take 180 degrees and subtract the other two angles from that. That would be what’s left for angle 𝐹𝐶𝐺. So angle 𝐹𝐶𝐺 will be equal to 180 minus two 𝑦 and the 𝑦. And simplifying, we could say angle 𝐹𝐶𝐺 will be equal to 180 minus three 𝑦.
Alright, now that we’ve labelled on pretty much every angle in our diagram, we’re trying to prove that 𝑥 is equal to three 𝑦 and we have an 𝑥 in the diagram that’s pink and an 𝑥 in the diagram that’s blue. Now, the 𝑥 in the diagram that is blue is located near a point with angles around that point. And angles around a point sum to 360 degrees. So if this is our point, all of the angles around this would sum to 360 degrees.
So if we wanted to come up with an equation that involved 𝑥, we could say that 𝑥 is equal to 360 minus those other three angles. So we take 360 and we need to subtract 90, 180 minus three 𝑦, and 90. And again, this is because angles around a point sum to 360 degrees. So let’s begin simplifying.
Inside our brackets, 90 plus 180 plus 90 is 360. But we can’t forget about the minus three 𝑦. Now, expanding this minus sign, we have that 𝑥 is equal to 360 minus 360 plus three 𝑦. So the 360s cancel. And we’re left with 𝑥 is equal to three 𝑦, which is exactly what we were trying to prove so we would be finished.