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Proofs - Proving Triangles Congruent Video

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okay guys this lesson is all about proofs and I know some of you think that's a bad word but give it a chance if you could do last night's homework it's the exact same you're just writing it out so if you could do last night's homework you can do this so let's just review real quick are five ways to show the trying congruent so the first way is side side side side angle side angle side angle angle angle side and hypotenuse leg now when you're trying to show and do a proof to show that the angles are congruent you were going to use one of these five methods so work the problem out before you do the proof and then you just have to write it all down so if you're going to use side side what is that what works in that problem in your proof you need to show all three sides are congruent in your proof you need to have down listed okay now some of it may already be in your given text counts and then you need to show the rest so if you're going to do side angle side you need to show in your proof two sides are congruent and  the included angle is congruent and if you're going to do angle side angle you need to show the two angles can rent and the included side is congruent okay now if you're going to do angle angle side which is different from angle side angle remember this side has to be between the two angles side has to be not between them okay so you need to show in your proof two angles that congruent and one have the non included sides congruent  so those three things must be in your proof somewhere you have to show this angles are congruent this set of angles are congruent and our sides are congruent and then you can say the triangles are congruent by angle angle side and then hypotenuse leg you need to make sure that you have right strangles first and then just show that the hypotenuse is  how to run out of here  are congruent  and one set of leg are congruent okay so in this one it's even easier you show the hypotenuse is or congruent you should the legs are congruent boom you're done the triangles are congruent by HL  okay here's two problems similar to what you had last night number one you have a given DB bisects angle ABC angle ABC by 6 this whole angle you know if it bisects it cuts it in half so this thing is going to be congruent to this angle and why is that because it bisects it okay and then they give you ABN BD are congruent so Abby and  BD are congruent that is a typo that should say PC Fix that on your paper BC are congruent so that's your give it will right now you already have a set of angles should have shot does anything else have to be congruent in this picture between this triangle and this track yes they share this site so now what do we have side angle side side angle side so yes these are congruent  by side side that's what you did last night now if you were going to write a proof of this  you've already done all the work you write down this information what's your given right there is your this set of sides are equal so there's one side then you write down this angles congruent to this angle why well because it bisects it and then the last thing would be to write down BD is congruent to BD why when they share a side that's the reflexive property you're done your last would be the triangles are congruent by SAS and that's your proof so let's look at number to okay given e is the midpoint of ab and a g is congruent to be h e is the midpoint of ab so if it's the midpoint what does it do cut in half so this is equal to this okay then Ag and BH are congruent so AG nvh will make those two marks and then that's everything in the given look at your picture does anything else have to be equal Cher side didn't vertical angles they do have vertical angles right there so now we have aside aside and angle side a side and an angle but does that work no it does not because side side angle does not work okay unless it is right triangle and it's not  okay now let's do a proof number one it tells me that BD is perpendicular to AC and AD is congruent to CD okay now those are already marked they won't all bmart but you can Mark they've already marked AD and DC and BD is perpendicular to AC know what they don't have marked is that's perfect where is that side is 90 what do you know about that side fit is 90 also okay and those are 90 because they're ridiculous right and then these two marks are given what else has to be true in that picture Beacon grunch they share site so BD is can run to BD now why is that true that is true because of the reflexive property so now I have side angle side side angle side they match up so siding side is why they are congruent now all you have to do is write it all down so statement number one is always your given guys if you can't do you can write down the given and I noticed on some test you haven't even written down the given and that's ridiculous because you can at least write that all you got to do is copy it okay so there's are given number two  let's look at what we did on our what we said these were both 90k and if they're both 9 either equal now all we have right now is that they're perpendicular we need to stay in our proof for the a that those angles are congruent so here's what I would do I would say first of all that angle be dije equals 90 degrees and angle B D C equals 90 degrees BDA BDC now why are they both 90° what was in the given that told you that they were 90° their perpendicular definition of perpendicular if they both equal 90 Breeze what do you know about him this equals 90 this equals 9 if they both equal 90 then they are congruent so right angle b d a is congruent to b d c k y they both equal. You could do substitution or transitive property here  turn transitive if they both equal the same Walmart so right now here is this side right here here is our angle which is right here so the last thing we have left do we need to write down this side will be D Tubidy  and why is BD congruent to BD what says anything is equal to itself the reflexive property okay and now I have my other side so I have my side my angle in my side everything's in your proof now the triangles are congruent  maybe. Just 10  Smurfette triangle ABC is congruent to Triangle C sorry BD by side angle side I remember this last statement has to be this proof statement exactly it has to be in this same order as the proof statement over to don't worry about what this says right here we don't need that okay so here's our picture here's our given B is the midpoint of so B is the midpoint of CD a b and e b Regal that's marked a b n e b and Anglesey India right angles okay so first thing we need to talk about B is the midpoint of CD if B is the midpoint of CD it cuts it in half so that is equal to well what cuz you're pretty much already done you have a side a side and an angle but that doesn't work side side angle but it's a right triangle so we have here we have a hypotenuse and a leg so this is my hypotenuse in this is my leg this is my hypotenuse and this is my leg  and you're done with so number one first thing you're given B is the midpoint  of needy  AB is congruent to EB and Anglesey angle Dr right  hello Sorry my phone's messing up a little bit and that is the given number to look at this and what do we need to write we already have these the hypotenuse we already have it's right there there's RH what do we need we just need those legs to the equal but we do need to establish that these are right triangles so we do need to say what we already have that don't we right here CD Sandy are both right angles so we just need to say that these legs are equal so we can say that CD is congruent to BD and why is that true because it told us it was the midpoint so that would be definition of midpoint  k now we have our leg so we have our hypotenuse and we have our leg we're done number 3 triangle ABC is congruent to Triangle EBD and that was by hypotenuse leg okay number 3 let's look at it work it out before you start writing that proof down it says AC bisects angle dab let's dab so it's this angle right here and AC bisects at with that bisects that we know that those two angles are congruent and why because it bisects so it'd be the definition of bisect okay and then it tells us it also by 6 DCB DCB so that's this single right here so now we know that this little is congruent to this angle and why is that because it bisects so have the same reason okay now we have an angle we're here in an angle pair here what else can be true oh look they share a site and remember that is always the search property  so now we have a side in between the two angles here's the two angles and the side is included so my reason this would be angle-side-angle now we just need to write each of these three things down and then it's done by angles okay so first thing always the given so number 1 AC  bisects single dab and angle DCB and that is given okay then we need to write down those two angles are congruent these two angles ignorant and this side they share is congruent it honestly does it matter what order you write it in if you want to do it an angle side angle form you can but you don't have to even do this angled in this angle and then the side and you can still use angle side angle because you just have to have all three parts in your proof so let's start with this angle up here so number two right angle in this is going to be b a c and then C A D or equal so it'll be be AC is congruent to angle  Cady is that well we put because it sex so that would be definition of bisects  now if you want to use that same reason because this is the exact same reason you can go ahead and do these two angles in the same step and use the same reason for that so let's do that and that's going to be can a CD so angle BCA is congruent to angle a CD and it's the same thing so step number 3 we need to do this side so AC is congruent to AC and that would be reflexive property again anything is equal to itself okay now if you'll notice here's our angle here's our angle in here's our side so we have all three switch since we have all three parts now we can write that the triangles are congruent triangle ABC is congruent to Triangle DC remember I'm just writing this down right here in the same order and my reason for that is angle side angle and you're done okay and the last one number for here's our triangle BD is perpendicular to AC so BD is perpendicular they see make those are perpendicular what do we know we know these two angles are 90° okay and that is because definition of K and then they tell us that a thing laying Angle serial so these two are equal so go ahead and Mark that and that's start of your given correct so you don't have to put that anywhere we already know that now what else has to be true in here do we have any vertical angles to the Shoreline that kind of stuff so if you look they share a line which many of these do so always look for that they share and the reason for that is reflexive so if you look we already have an angle these two are equal because they're both 90 we're going to have establish that in her proof and we have a side so angle angle side angle angle side so we're going to prove it by angle site so now all you're going to dis is your given so you just need to write down these two are equal because they're perpendicular and you need to write down these or equal because of reflexive and so let's do our little grid statements reasons number 1 BD is perpendicular AC and angle a is congruent to angle C that's given number two now  all are given till this is their perpendicular are given does not tell us they are both 90 and that makes them equal so we need to establish because they're perpendicular they're both 90 and because they're both 90 their equal so we do have to use those couple of steps you got to do it step-by-step I know you know true but you got to prove it's true so you can say angle ADB equals 90 degrees single BDC equals 90 degrees number three since they both equal 90 then they are congruent to each other correct so angle ADB is can put two angle BDC these are both 90 because they were perpendicular so that's definition of perpendicular  now they are equal to each other because they both equal 90 they have to be people show that sets up transitive or substitution whichever you would like to do there is also a theorem and I think it's 2.3 that says all when was our King you could use that too if you want to okay so now we have  right here we have one of our angles in right here we have our angle what are we missing we're missing that side so let's write down that side next  look for peanut side is BD that they share so BD is congruent to BD and that is reflexive property  came in so there's our site so now we have our angle or England our side in our proof and again they you have to be in that order you just have to have all three parts but it still is angle angle side so number five  you know we're going to write this exactly like it is triangle ABC is congruent to Triangle CBD and that is by angle angle side your homework is all proves there's like 9 of them I'm going to give you a hint the last couple or a little bit harder than the rest there is one or two that you need to use the segment addition postulate one side of the triangle is going to be in two parts you're going to have to add the two parts to get the whole side of the triangle and then you're going to use substitution or transitive or something like that to show that they're equal to the other one in the blue triangle so I want you to think about that if you need help come see me the other reason that I did it this way  was that you can go back and watch this again and again if you need to rewind rewatch until it clicks if you need help just come see me

Uploaded by Pam Chastain.

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