### Video Transcript

In the diagram, π΄π΅πΆπ· is a kite, and π΅πΈπΉπΆ and πΆπΊπ»π· are squares. Prove that π₯ is equal to three π¦.

So our goal is to prove that π₯ is equal to three π¦. And we have two π¦ and π¦ in our diagram, along with our angle π₯. So to prove this, letβs first go through the given information and try to use properties that might be useful to prove this.

So first, we know that π΄π΅πΆπ· is a kite. So if we have a kite, we can say that angle π΅π΄π· is equal to angle π΅πΆπ· because itβs a kite. In a kite, the angles across from each other are congruent. So like we said, angle π΅π΄π· and angle π΅πΆπ· would be equal. And since angle π΅π΄π· is equal to π₯, we can also call angle π΅πΆπ· π₯.

Next, we are given two squares: π΅πΈπΉπΆ and πΆπΊπ»π·. Now instead of using every property of a square and every property of a kite to label this diagram, weβre trying to prove that π₯ is equal to three π¦ and all of the π₯s and π¦s that are found in our diagram are angles. So letβs 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 theyβre all equal to each other and each of them will be 90 degrees because π΅πΈπΉπΆ is a square. And because πΆπΊπ»π· is a square, each of its angles are equal to 90 degrees.

So the only shape that we havenβt touched is triangle πΆπΉπΊ. But we were given that angle πΆπΉπΊ was equal to two π¦ and angle πΉπΊπΆ is equal to π¦. And all angles in a triangle sum to 180 degrees. So this missing angle πΉπΆπΊ, we could say, we take 180 degrees and subtract the other two angles from that. That would be whatβs left for angle πΉπΆπΊ. So angle πΉπΆπΊ will be equal to 180 minus two π¦ and the π¦. And simplifying, we could say angle πΉπΆπΊ will be equal to 180 minus three π¦.

Alright, now that weβve labelled on pretty much every angle in our diagram, weβre trying to prove that π₯ is equal to three π¦ and we have an π₯ in the diagram thatβs pink and an π₯ in the diagram thatβs blue. Now, the π₯ 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 π₯, we could say that π₯ is equal to 360 minus those other three angles. So we take 360 and we need to subtract 90, 180 minus three π¦, and 90. And again, this is because angles around a point sum to 360 degrees. So letβs begin simplifying.

Inside our brackets, 90 plus 180 plus 90 is 360. But we canβt forget about the minus three π¦. Now, expanding this minus sign, we have that π₯ is equal to 360 minus 360 plus three π¦. So the 360s cancel. And weβre left with π₯ is equal to three π¦, which is exactly what we were trying to prove so we would be finished.