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Proof: The diagonals of a kite are perpendicular

Sal proves that the diagonals of a kite are perpendicular, by using the SSS and SAS triangle congruence criteria. Created by Sal Khan.

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Video transcript

What I want to do in this video is I want to prove that segment AC is perpendicular to segment DB based on the information that we have in this diagram over here. That this side has the same length as that side, this side has the same length as that side. And I'll give you a hint, we're going to use one or more of our congruence postulates. I'll just stick with calling them postulates from now on. And so the ones that we know-- so let me draw a little line here. This is kind of our tool kit. We have the side side side postulate, if the three sides are congruent, then the two triangles are congruent. We have side angle side, two sides and the angle in between are congruent, then the two triangles are congruent. We have ASA, two angles with a side in between. And then we have AAS, two angles and then a side. So any of these things we've established-- these are our postulates-- we're going to assume that they imply congruency. And I'm also going to do this as what we call a two column proof. And you don't have to do something as a two column proof, but this is what you normally see in a normal introductory geometry class. So I thought I would expose you to it. It's a pretty basic idea is that you make a statement, and you just have to give the reason for your statement. Which is what we've been doing with any proof, but we haven't always put it in a very structured way. So I'm just going to do it like this. I'll have two columns like that. And I'll have a statement, and then I will give the reason for the statement. And so the strategy that I'm going to try to do is it looks like, right off the bat, it seems like I can prove the triangle CDA is congruent to triangle CBA based on side side side. And that's a pretty good starting point because once I can base congruency, then I can start to have angles be the same. And the reason why I can do that is because this side is the same as that side, this side the same as that side, and they both share that side. But I don't want to just do it verbally this time, I want to write it out properly early in this two column proof. So we have CD, we had the length of segment CD is equal to the length of CB. CD is equal to CB, and that is given. So these two characters have the same length. We also know that DA, the length of segment DA, is the same as the length of segment BA. So DA is equal to BA, that's also given in the diagram. And then we also know that CA is equal to CA, I guess we could say. So CA is equal to itself. And it's obviously in both triangles. So this is also given, or it's obvious from the diagram. It's a bit obvious. Both triangles share that side. So we have two triangles. Their corresponding sides have the same length, and so we know that they're congruent. So we know that triangle CDA is congruent to triangle CBA. And we know that by the side side side postulate and the statements given up here. Actually, let me number our statements just so we can refer back to this 1, 2, 3, and 4. And so side side side postulate and 1, 2, and 3-- statements 1, 2, and 3. So statements 1, 2, and 3 and the side side postulate let us know that these two triangles are congruent. And then if these are congruent, then we know, for example, we know that all of their corresponding angles are equivalent. So for example, this angle is going to be equal to that angle. So let's make that statement right over there. We know that angle DCE-- so this is going to be statement 5-- we know that angle DCE, that's this angle right over here, is going to have the same measure, we could even say they're congruent. I'll say the measure of angle DCE is going to be equal to the measure of angle BCE. And this comes straight out of statement 4, congruency, I could put it in parentheses. Congruency of those triangles. This implies straight, because they're both part of this larger triangle, they are the corresponding angles, so they're going to have the exact same measure. Now it seems like we could do something pretty interesting with these two smaller triangles at the top left and the top right of this, looks like, a kite like figure. Because we have a side, two corresponding sides are congruent, two corresponding angles are congruent, and they have a side in common. They have this side in common right over here. So let's first just establish that they have this side in common right over here. So I'll just write statement 6. We have CE, the measure or the length of that line, is equal to itself. Once again, this is just obvious. It's the same. Obvious from diagram it's the same line. Obvious from diagram. But now we can use that information. So we don't have three sides, we haven't proven to ourselves that this side is the same as this side, that DE has the same length as EB. But we do have a side, an angle between the sides, and then another side. And so this looks pretty interesting for our side angle side postulate. And so we can say, by the side angle side postulate, we can say that triangle DCE is congruent to triangle BCE. And when I write the labels for the triangles, I'm making sure that I'm kind of putting the corresponding point. So I started at D, then went to C, then to E. So the corresponding I guess angle, or the corresponding point or vertex I could say, for this triangle right over here, is B. So if I start with D, I start with B. C in the middle is the corresponding vertex for either of these triangles, so I put it in the middle. And then they both go to E. And that's just to make sure that we are specifying what's corresponding to what. And we know this, we know this is true, by side angle side. And the information we got from-- so we got this side is established that these two sides are congruent was from statement 1. Then that these angles are congruent is from statement 5 right over here. And then statement 6 gave us the other side. Statement 6, just like that. And if we know that these triangles are congruent, that means that all of their corresponding angles are congruent. So we know, for example, that this angle right over here is going to be congruent to that angle over there. So let's write that down. So we know, statement number 8, the measure of angle, let's call that DEC, is equal to the measure of angle BEC. And this comes straight from statement 7. Once again, they're congruent. Congruency. And then we also know-- we'll make statement 9. We also know that the measure of angle DEC-- or maybe we should just write it this way. Angle DEC and angle BEC are supplementary. They are supplementary, and that's kind of-- you can just look at that from inspection but I'll write a decent-- are supplementary. Supplementary, which means they add up, their measures add up, to 180 degrees. And we know that because they are adjacent and outer sides form straight angle. And then, the next step-- if we know that these two angles are equal to each other, and if we know that they are complementary-- our next step means that we could actually deduce that they must be 90 degrees. So 10, measure of angle DEC equals measure of angle BEC, which equals 90 degrees. And then for the reason it might be a little bit more involved, we could put these two statements together. So it would be statements 8 and 9. And then statements 8 and 9 mean that DEC, so I could write this, measure of angle DEC plus measure of angle-- actually, let me just, since I don't want to do too many steps all at once, let me just take it little bit by little bit. So let me just do it all like this. So let me say measure of angle DEC plus measure of angle BEC is equal to 180. And this comes straight from point 9, that they are supplementary. And then we could say statement-- I'm taking up a lot of space now-- statement 11, we could say measure of angle DEC plus measure of angle DEC is equal to 180 degrees. And we know that from statement 9. We know that from statement 9 and statement 8. We essentially just took statement 9 and substituted that BEC, the measure of BEC, is the same as the measure of DEC. And so then, if we want statement 12, we could say measure of angle DEC is equal to 90, which is equal to the measure of angle BEC. And then, this comes once again straight out of point number 11 and 8. And what you could see, I'm kind of taking a little bit more time, going a little bit more granular through the steps. In some of the other proofs I would've just said, oh, obviously this implies this or that. And then we're done. Because if these are 90 degrees-- so let me write the last statement. So statement 13, which is what we wanted to prove. We wanted to prove that AC is perpendicular to DB. So AC is perpendicular to, what was it? AC is perpendicular to segment DB, and it comes straight out of point 12. And we're done. We've done a two column proof, and we have proven that this line segment right over here is perpendicular to that line segment right over there. We did it with SSS, with the SSS postulate, and the side angle side postulate.