in this video you're going to learn how to do two column proofs involving proving triangles congruent and we're going to go through five different proofs together we're going to do one of each of the different types side side side side angle side angle side angle angle angle side and hypotenuse leg and if you've been having trouble understanding this concept I'm going to show you some tips and techniques as we go through to make this a little bit easier so let's dive into the first example we're given that AB is parallel to CD Now whenever you think of the Givens think of them as clues or hints and take the moment to just write this on the diagram so that you don't have to remember it you can kind of visualize it or see it and that's what I'm doing with these arrows I'm showing those are parallel and ab is congruent to CD so I'm just going to put the corresponding little Dash there to show that they're the same length we want to prove that the triangles are congruent but again think of the Givens as clues or hints like why are they telling us that these lines are parallel well you remember probably from a earlier chapter parallel lines okay I like to extend them a little bit when they're parallel like this cut by a transversal what do we end up having well we end up having angles that are congruent and in this particular problem you can see that this angle and this angle are congruent by alternate interior angles right so they're in between the two parallel lines one's on the right one's on the left side of the transversal and they're going to be congruent to each other when you figure something out you're going to want to put that as a step in your proof but normally what we do is we start with writing the Givens as our first step you don't have to sometimes people will scatter them a little bit through the proof as they go but most students what they'll do is they'll just list these Givens right here in the first step so I'm going to say AB is parallel to CD okay and also um we'll say AB is congruent to CD and we're just going to say that that's given but for step number two we figured out that these angles were congruent by alternate interior angles I'll just abbreviate AIA for alternate interior angles and notice that there's more than one angle at this vertex so I'm going to say angle DCA is congruent to angle B A okay now let's look at what we have here we've got this side and this side are congruent these angles are congruent and then notice that these two triangles they share this side in between them AC so AC is going to be congruent to AC by the reflexive property remember it's just like looking in a mirror your reflection in the mirror like you're the same height in the mirror as you are in real in real life so we call that the reflexive property I'll just sit right reflexive I'm just going to abbreviate now do we have enough information to prove these two triangles congruent now remember triangles are actually consist of six pieces of information you have three sides and three angles but by showing these specific components are congruent that's enough to show that the whole triangle is congruent and we only really need three pieces and in this case you can see the angle is in between the two sides that's called the included angle and that would be the side angle side so this is proving the triangle is congruent by side angle side and remember whatever you're trying to prove that's going to be the last step in your proof here so this is going to be step number four here triangle ABC is congruent to Triangle CDA now another important thing when you're looking at these triangle congruence statement it tells you how things should match up like a right should match up side CD see first and second first and second or BC should match up with da or AC should match up with CA Etc let's take a look at another example okay for example number two try this one if you feel like you're getting the hang of this we're given that and I'm just abbreviating and G forgiven D is the midpoint of AC now why is that a clue or a hint well midpoint means it's in the middle right so that means that this length should be the same length as that length so let's put that as a step in our proof now I'm going to jump down to step number two step number one we already mentioned that that's going to be our given right so we can just draw a little arrow here for the given but step number two we can see that a d is congruent to CD a d is congruent to C CD and that's the definition I'm just going to abbreviate of midpoint abbreviating okay now notice when I figure something out I like to mark it on the diagram again so I don't have to remember all that information they also tell us that ba is congruent to BC So ba is congruent to this length BC again marking it on the diagram so I don't have to remember it that's in our given now anything else that's congruent between the two triangles any sides or any angles there's a couple different ways to do this problem I mean one way to do it would be to say that oh I can see that this is an isoceles triangle and the base angles are congruent that's one option but here I think I'm just going to take the easy rote here by noticing that this side is shared between these two triangles it's kind of like if you go back to back with a friend say oh we're the same height because you're back to back see how the triangles are are right next to each other so we know that ad is congruent to itself a d and you'll notice that this comes up a lot in proofs the reflexive property anything is going to be congruent to itself now do we have enough information to show these triangles are congruent yes we've got three sides are congruent to three sides that's our side side side and so that's going to be our last step number four remember whatever you're trying to prove goes in the last step and the reason is SSS or side side side let's take a look at another example okay see if you can do number three now we've got uh our given that angle a is congruent to angle e so let's mark that on the diagram these angles are the same and we've got segment AE bcts uh segment BD now what does that mean that AE is bisecting segment BD well we know I means two SE means to like intersect or cut so which segment is cutting which segment well AE is the segment that's cutting BD in half so that tells us that BC is congruent to CD let's put that as a step in our proof so when you figure something out again the clues are uh in the Givens there and we can sometimes it's just a gimme like they just kind of gave us this information which I marked so that's going to go in Step number one all of our Givens but for step number two we figured out that BC is congruent to DC and that's by the definition of segment bis sector okay I'm just abbreviating so a lot of times you'll notice that the reasons they're going to be theorems but they're also going to be definitions and it's just definition of segment bis sector the last one we had a definition of midpoint definition of supplementary angles a lot of times you'll see that it's just the definition now is there anything else that's congruent between these two triangles any sides or any angles well notice that whenever you see like an X like this where two lines intersect the angles that are across from each other those are going to be congruent those are called vertical angles so what we're going to do here for step number three is we're going to say angle ACB is congruent to angle uh let's see what is this ecd and that's by the vertical angle congruence theorem I'm just going to abbreviate VA vertical angles sometimes students will say oh uh angle C is equal to angle C but notice when there's more than one angle at a given vertex at a given point to be clear you want to use three points make sure that middle letter is the hinge or the vertex of the angle now do we have enough information to prove that these two triangles are congruent using one of these over here on the right well it looks like we've got angle angle side angle angle side it's like if I was to take this triangle and rotate it those components would match up with one another and that's enough to prove that the entire uh triangles congr to the entire triangle now when you do these uh okay you want go around the triangle you want to go like in order like angle the next angle you come to the next side you come to you don't want to do like some random like angle you know what I mean you go around in order clockwise or counterclockwise okay um you don't want to skip around okay so now we said that the triangles are congruent that's going to be our last step in our proof what we're trying to prove and we said that was by angle angle side and you got it let's take a look at another example see if you can do example number four we've got BD is perpendicular to AC and BD bcts angle ABC so why are those clues or hints that's what we want to figure out remember our first step we always want to list our our Givens okay that's usually uh common practice amongst a lot of students just to kind of get that out of the way sometimes it is helpful to do them one at a time like if you're addressing this given you might want to put that first and then what you kind of derive from that and then take the next given what you derive from that but for this one we're just going to put the Givens in our first step uh for number two now why do they tell us BD is perpendicular to AC well we know when it's perpendicular it's going to form a right angle right and so let's go ahead and write that down so we're going to say uh angle a DB is a right angle I'm going to abbreviate and angle C DB is a right angle and the reason for that again another definition it's it's the definition of perpendicular I'm just abbreviating now we can take this one step further and we can say now that angle a DB is congruent to angle C DB and that's by the right angle congruence theorem meaning that all right angles are congruent to all of the right angles they call that the right angle congruence theorem now taking this other given that BD segment BD bcts angle ABC what does that mean well we talked about how by means two SEC means cut but it's cutting an angle now in half this top angle ABC so if it bcts it that tells us that these two guys are going to be congruent to one another and we want to list that as a step in our proof whatever we figure out so angle a b d is congruent to angle c b d and the reason is definition of angle I'm just abbreviating bis sector okay now is there anything else that's congruent between these two triangles well again you can see that BD is shared between these two triangles so by the reflexive property we know that BD segment BD is congruent to segment BD and that is reflexive just like looking at a mirror you're the same height in real life as you are in the mirror now do we have enough to prove that the two triangles are congruent that's our goal that's what we're trying to prove well we can see we have angle side angle the side this is called the included side it's in between these two angles that's this guy right here angle side angle so let's go ahead and make that what we're trying to prove the last step in our proof right here and the reason we abbreviate ASA angle side angle let's take a look at one last example okay so for this last example try it if you want to uh get some practice but we're given this angle ABC ab C is a right angle so let's mark that and we're told that angle DBC is a right angle so oh I meant for this to be DCB my mistake DCB is a right angle that's this guy right here okay so we'll mark that we also know that AC is congruent to BD so let's see AC is congruent to BD now what's interesting about this problem we want to prove these two triangles are congruent triangle ABC and triangle DCB that's this big triangle here and this big triangle here they're overlapping and so what I oftentimes recommend to students when they're overlapping is to pull them apart and draw them separately and that's what I did down here it's a little bit easier to analyze when they're like that and we can Mark these as well so this is a right angle and this is a right angle that was given and they also told us that AC was congruent to BD okay so that's interesting now let's go to our our proof we know we're putting our given here in the first step say given but step number two you see how these angles are both right angles so let's say that angle AB C is congruent to angle DCB and the reason is um right angle congruence theorem right right angle congruence theorem all right angles are congruent to all the right angles we could also state in Step number three that triangle ABC triangle ABC is a right triangle in and triangle uh DCB is a right triangle and that's the definition of a right triangle if it has a right angle uh then that's a right triangle right so now what we're going to do is we're going to say that uh and this is where students kind of Miss see because the two triangles are overlapping if I pull them apart see how this is BC and BC they originally shared that side so BC is going to be congruent to itself s by the good old fashioned reflexive property right we've seen that a lot today in these proofs so BC is congruent to BC reflexive sometimes students miss this one because of it's hard to kind of notice that it's not like the two triangles are side by side they're actually overlapping at that that side there and now do we have enough information to prove that these two triangles are congruent well we know that they're right triangles we also know that the hypotenuse is diagonally across from that right angle and those are congruent and we have one leg that's congruent to one leg that's what we call our hypotenuse leg theorem it looks at first glance like there's only two components H and L like hypotenuse and leg but in order for it to have a hypotenuse it also has to be a right triangle or have that right angle so there really is three components here uh remember with these triangles you cannot prove triangles congruent by angle angle angle uh there's no angle side side or side side angle these are not ways to prove triangles congruent just these five but in this last example what we're trying to prove is going to be our last step and we said that that was because of the hypotenuse leg theorem so great job if you're able to follow all these uh triangle congruence proofs if you want more practice I'll try to put a playlist right there with some more triangle congruence to column proo like this so you can get some additional practice I'll see you over in those videos