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Current time:0:00Total duration:11:06

Proof: The diagonals of a kite are perpendicular

CCSS.Math:

Video transcript

what I want to do in this video is I want to prove 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 it 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 of our congruence postulates I'll call them I'll just stick with calling on postulates from now on and so the ones that we know so let me draw a little line here this is kind of our toolkit 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 a sa two angles with a side in between and then we have AAS two angles and then aside so any of these things we've established these are our postulates we're going to assume that they imply congruence 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 normally an introductory geometry class so I thought I would expose you to it but this it's a pretty 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 don't always 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 two columns write this like that and I'll have a statement I'll have a statement and then I will give the reason 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 I can like it seems like I can prove that 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 congruence 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 that I don't want to just do it verbally this time I want to write it out properly in this two column proof so we have CD we had the length of segment CD is equal to the length of C B C D is equal to C B and that is given that is given so these two characters have the same length we also know that D a this the length of segment D a is the same as the length of segment B a so da da is equal to BA that's also given given in the diagram and then we also know that C a is equal to AC a I guess we could say so C a is equal to itself and it's obviously in both triangles so this is also given or it's obvious from the diagram or obvious from the diagram it's a bit obvious both triangles share that side so we have we have two triangles they have their corresponding sides have the same length and so we know that they are congruent so we know that triangle triangle C D a triangle CDA is congruent to triangle CBA to C B CBA and we know that by the side-side-side postulate and the statements given up here and let actually let me number our statements just so we can refer back to this one two three and four and so side-side-side postulate and one two and three statements one two and three so statements one two and three 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 we know that 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 D C e that's this angle right over here is going to have the same measure we can even say they're congruent I'll say the measure of angle D see E is going to be equal to the measure of angle of BCE B and this comes straight out of statement for this comes straight out of statement for congruence I could put in parenthesis congruence of those triangles this implies straight because they're both part of this larger triangle they are the corresponding angles so they are going to have the exact same measure now it seems like we can do something pretty interesting with these two smaller triangles at the kind of 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 six we have c e the measure or the length of that of that line is equal to itself once again this is just obvious it's the same it's the same obvious from diagram it's the same line obvious from 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 pretty interesting for our side angle side postulate and so we can say we can say by the side angle side postulate we can say that triangle DCE triangle DC triangle DC e DC e is congruent is congruent to triangle to triangle BCE and when I write the labels for the triangles I'm making sure that I'm kind of putting the corresponding points 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 for this triangle right over here is B so I started if I start with D I started with B C in the middle it's the corresponding vertex for either of these triangle so I put 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 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 one then that angles that these angles are congruent is from statement five right over here and then statement six gave us the other side statement six 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 eight the measure of angle is called that Dec Dec is equal to the measure of angle B EC b ec b EC and this comes straight from statement this comes straight from statement seven once again they're congruent congruence congruence and then we also know will make a statement nine we also know that the measure of angle Dec or maybe we should just write this way angle angle Dec and angle B EC 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 they're adjacent and outer sides outer sides form straight angle form straight angle straight angle straight angle and then we can essentially 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 can we've actually deduced that they must be ninety degrees so ten measure of angle Dec equals measure of angle B EC which equals ninety which equals ninety 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 of angle actually let me just since I want to I don't want to do too too many steps all at once let me just take it a little bit by a little bit so let me just do it all like this so let's say measure of angle Dec plus measure of angle B EC is equal to 180 and this comes straight from 0.9 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 nine we know that from statement nine and statement eight and statement eight we essentially just took statement nine and substituted that Dec the measure of B EC is the same as a measure of Dec and so then if we want to statement twelve we could say a measure of angle Dec is equal to 90 which is equal to the measure of angle B EC measure of angle B EC and then this comes once again straight out of straight out of point number 11 and eight and what you can see I'm kind of taking a little bit more time going a little bit more granular through the steps and some of the other proofs I didn't said Oh obviously this implies this or that and then we're done because if these are 90 degrees 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 D be AC is perpendicular to D B so AC is perpendicular to was it to to a C's particular DB to segment DB and it's comes straight out of point 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 the side-angle-side postulate