hi everyone in this video we're going to be taking a look at congruent triangle proofs but ones that have quadrilaterals involved in them so sometimes quadrilaterals are incorporated into a typical triangle congruency proof these proofs basically require us to use the properties of quadrilaterals to help prove whatever the desired outcome is so first let's recap um a standard triangle congruency proof and what those steps are so I've listed them here first we always start by writing out whatever our given statements are then we look and say can we elaborate upon them are there any keywords or symbols in the Givens that we could now talk a little bit more about we can look to see if the reflexive property can be used that's any time the triangles share a side or angle we're going to look for vertical angles those are angles that are congruent and formed by intersecting lines finally we would prove the triangles is congruent and then determine if CP CTC applies so that's a recap of the basic steps of how to do a congruent triangle proof but all the proofs we're going to look at in this video are going to relate to quadrilaterals so parallelograms rectangles run by and squares are the most common quadrilaterals seen in proofs we're going to look at the space below to list the properties of these shapes and for the sake of the video I've already gone ahead and filled these in so some key facts about a parallelogram opposite sides are congruent opposite sides are parallel opposite angles are congruent the consecutive angles are supplementary and the diagonals bisect the ankles over on the right a rectangle is a type of parallelogram that means it inherits all of the parallelogram properties but there's some new properties of a of a rectangle which is that the consecutive sides are perpendicular that's how we get the right angles in the corners and the diagonals are now congruent right in our bottom row here a rhombus is a type of parallelogram so it also holds its properties in addition all the sides of a rhombus are congruent the diagonals of a rhombus are perpendicular and the diagonals bisect the angles last square is basically a type of parallelogram it's a type of rectangle it's a type of rhombus so it inherits the properties of all of those so we're going to look at a couple of proofs in this video and depending what shape they get us let's say they give us a parallelogram we're going to need to use one of these or more one or more of these parallelogram properties all right let's take a look at our first example and if you need to pause the video to write anything down at any point of course feel free to do so so question one given rectangle ABC D with diagonal AC is shown and we're going to prove that triangle ADC and triangle ABC are congruent so I have already gone ahead and done my first step of writing the given statement so I set up my two column proof with statement and reason I wrote my Givens and now I'm ready to move on to the next step of a proof one thing that's also important to note here is that when you're doing proofs with quadrilaterals it's very common for there to be multiple correct Solutions so I'll just be walking through one in the video uh but it is possible that other options exist all right so I know that we're working with a rectangle so I have to think about well what facts do I know about a rectangle and one that I know is that opposite sides are congruent so that would mean that a d and BC are congruent and ab and CD are congruent so I'm going to Mark those off in my diagram and let's write that in here so I know that a d is congruent to CB and I know that AB is congruent to CD for how we know that well let's think about what property we just used so I'm going to say in a rectangle the opposite sides are congruent this is an example of us elaborating upon the Givens we looked at the Givens for any keywords in this case rectangle and we talked more about what it means to be a rectangle next we will look at the reflexive property so both of these triangles share side AC so I'm going to say that AC is congruent to AC by the reflexive property and I'm going to mark that in my picture when we are looking at this here the two triangles now have three markings that means generally I can prove they're congruent so I'm going to write my proof statement triangle ADC is congruent to Triangle AB BC and I have side side side congruency here other things you could have done in this proof you could have talked about the right angles instead of the reflexive property that would give you side angle side that's another option um there's definitely other things you could do alternate interior angles I did kind of what I thought was the quickest and most straightforward method for this proof all right let's take a look at another and again this proof has multiple Solutions I'm going to pick what I I feel is the most straightforward one all right we're given parallelogram ABC D diagonals AC and BD intersect at e and we want to prove triangle a d and CB are congruent so there's a few different triangles in this picture so let's just take note of the ones that we're trying to prove are congruent to one another all right so based on that I've already gone ahead wrote my given in here um I'm going to actually start off using the same fact I did in um the previous question which is that opposite sides are congruent so I'm going to say that a d is congruent to CB because opposite sides of a parallelogram are congruent you certainly could mention a b and CD if you wanted but those are not part of the triangles that I'm trying to prove congruent so I'm just going to leave that information out all right I'm still thinking about this being a parallelogram and what else that means so one fact that I know about parallelograms and again just a reminder here's your reference here as to what facts you can use but another fact I know is that the diagonals BCT each other so I'm going to write that in here I'm going to say that AC and BD BCT each other and my reason will be in a parallelogram the diagonals bisect each other now let's think about that word bisect right in a proof you're always kind of like building off of a previous fact BCT means that if a line segment is bisected it's split into two congruent parts so for instance I know that AE is congruent to CE I also know that de is congruent to be and I'll mark that in my picture so AE and C D and be and the reason behind that is that I bis sector splits a segment into two congruent parts and now pretty similar to the last question I have three things marked off in each triangle I can now prove these two triangles are congruent that's my proof statement here and it's once again side side side congruency other options you could have done here is perhaps not talk about the opposite sides but do the vertical angles instead um and maybe have side angle side you could have talked about some alternate interior angles as well all right let's look at one last one here and this one's going to be a little bit more difficult I can kind of tell that this is set out to be more difficult because of the way the picture looks it doesn't just look like a rectangle or a parallelogram but there's kind of some additions here um not just diagonals either there's extra segments and looks like two extra triangles kind of hanging up sides um so we're going to have to really think about what to do on this proof a little bit more so we know that parallelogram uh or that abc e is a parallelogram and we also know that angle AF is congruent to BDC so I'm going to mark off those two angles in my picture and our goal is going to be yes we want to do our proof statement a is congruent to BD but we have to first get these triangles to be congruent to one another so since the proof statement is talking about corresponding parts of congruent triangles right AF is part of that triangle then I know that this is going to be a proof that requires cpctc but in order to use that you have to have congruent triangles first all right so let's talk about how we can figure out that these two triangles are congruent I'm going to reuse the same fact we've been using in this video I'm going to say that AE is congruent to BC because the opposite sides in a parallelogram are concurent and if you're a little stuck like looking at the picture and identifying the parallelogram grab a highlighter remember it says the parallelogram is a b c e and now you can see that a e and BC are the opposite sides and therefore they're congruent and I'm going to put tick marks on them so actually our proof is looking pretty good so far because those two triangles already have two markings we just have to find a way to get a third piece to be congruent so that we can pick one of our triangle congruency methods so I have nothing to elaborate upon here again besides that there's a parallelogram so I'm going to think about other facts of a parallelogram well I know in a parallelogram the opposite sides are congruent and we talked about that but the opposite sides are also parallel so I'm going to write that for line three I'm going to say that AE is parallel to BC and my reason is going to be the opposite sides of a parallelogram are parallel and we'll talk about how this is going to help us in a second so when you have parallel lines often in proofs that would indicate alternate interior angles that's the most common scenario um but there's another one that definitely shows up which is called corresponding angles corresponding angles are two angles that are formed by parallel lines in a transversal and they're congruent to one another they're in the same location so I'm going to just zoom in for a second here and draw you an example so you can see what I'm referring to let's say I have two parallel lines and let's say I have a transversal this angle and this angle are going to be congruent to one another because they're corresponding they're both in the same like little location in their set so like if I look here's four angles here's four angles those are both in like the top left so I actually have corresponding angles in this diagram here angle a e f and angle b c d are corresponding angles because here's our parallel lines AE and BC we just said that and then FC is our transversal cutting through it so let's write about these corresponding angles so I'm going to say that um angle a EF and angle BCD are corresponding angles and the reason I know that is because parallel lines this is why we had to establish lines were parallel by the way parallel lines cut by a transversal form corresponding angles now corresponding angles are always congruent so I'm going to add an additional line in here talking about that I'm going to say those two angles are congruent because corresponding angles are congruent so now we have successfully marked off three pieces of each triangle I'm going to go back to my picture for a second I'm going to eliminate some of this pink highlight and now you can see in our two triangles there are three markings that gives us enough information to prove the triangles are congruent so I'm going to say for line six triangle a EF is congruent to Triangle BCD and in this case that's by angle angle side congruency and now finally I can do my proof statement if I have congruent triangles all of their corresponding pieces are now congruent so I can say that segment AF is congruent to segment BD by cbcc and we have completed this proof hopefully in this video you learned a little bit more about how to take parallelograms rectangles and other quadrilaterals from the Givens and how to talk about them within a proof