Alright, welcome. So this is 4.1, Congruent Figures. So make sure you have 4.1, Congruent Figures, at the top of your paper. So our objective today is to recognize congruent figures and their corresponding parts. Okay?
Alright. So the first thing, you don't need to write this down, but let's take a look at this picture. Alright, so congruent figures are figures that have the same size and the same shape.
So you don't need to write this down, but if you look at this, if I were to slide the figure up here, The rectangle stays the same, right? The sides didn't get bigger, like the angles didn't change. It's still the same polygon. The same thing right here. If I flip that polygon right there, my sides and angles are still the same.
All I did was I flipped it. Same thing if I turn this triangle, my sides and all that stay the same. So that's kind of what we're dealing with in this concept today.
So you are going to write down this right here, this key concept. So make sure you have congruent figures at the top, and then write down everything inside the box, as well as our example and the congruent parts. So, as you're writing that down, congruent polygons have congruent corresponding parts. So that means that the segments are going to be the same. So we say that in this one right here, this segment of AB is corresponding part.
spot little the corresponding segments gotta know I was so hard to say is fe or EF right there so each of those segments are corresponding you see how they match up all we really did was we took that plug on and we flipped it also the angles here here angle a is also congruent to angle e so my angles are also congruent those are also the same So make sure you have all the angles and all the segments written down. Okay? Alright.
Alright, so we're going to write down this box right here and everything inside of it. So this is called our third angle theorem. So what it says, if two angles of one triangle are congruent to two angles of another triangle, then that means that the third angles are congruent. So, if I'm thinking about this, if angle A and D are congruent, and let's say they are both... 30 degrees and it tells me that B and E are congruent and let's say those are 50 degrees well we know that a triangle has to add up to the interior angles add up to 180 degrees so both of these are combined to be 80 we know that for both of these that they are going to equal 100 degrees so this is what the third angle is that both of these C and F would be congruent.
So that's what that theorem is, so make sure that you write everything down in it. Okay, let's go to some problems now. Alright, so let's do our problem number one right here. So before we go, I didn't really emphasize it here, but it said when we wrote this down, the one box we have, when you name congruent polygons, you must list corresponding vertices in the same order.
So we have A, B, C, D. It has to be, the corresponding spots have to be in the same order. So they'd be E, F, E, F, G, H. So what I'm going to do is I'm going to use some colors on this first one to show which angles. And remember when corresponding spots, corresponding parts, we have sides and angles.
Those are the two things that are going to be congruent. So we can... find the angles first.
So we'll highlight ones that are the same. So I see that H right here is going to be the same thing as L. Let's do red right here. I know that I is going to be the same thing as M.
I is going to be the same thing as M. And let's go sky blue. We have J is going to be the same thing. J right here is going to be the same thing as N. And our last one, let's get some green in there, yellow-green.
K is going to be the same thing as O right there. So those are the same angles. Now, I could have seen by the picture, seen how I rotated, and those are the same. But another way I can tell that they are the same is when I looked at here, when it told me corresponding spots, I see that H is the same thing as L. I is the same thing as M.
Those are the second ones. So all the angles that I have in common are these ones that I just highlighted. So I can say H is congruent. L, I is congruent to M, and J is congruent to N, and K is congruent to O. All the ones that I just highlighted.
Now that we see that these ones right here are congruent, I know that the segments, so if I'm talking from yellow to green, that would be HK, is going to be the same thing as yellow to green, O to L. So I can see the different corresponding spots as far as segments. And there are all my corresponding segments right there. So remember, when my figures are congruent, then I have the same sides and I have the same angles. And I list it.
A really easy way to tell is when you are first listing it, those have to be in the same order. Make sure you have that all written down. Alright, so our last problem that we're going to do right here. So we are going to try and... prove through our, for our proof table right here that these two triangles are congruent.
So we see right here that we have these two triangles and that they share this line right here. We could even separate these two triangles into two different things. So in order to prove that figures are congruent you need six things, six things. Three of them.
You need to have three angles that are the same. And you need to have three sides that are the same. If you can prove those six things, then the two figures are congruent. So what we're doing right here, we see that we already have a couple sides that are the same.
We have two sides, and we have two angles that are the same. So we've got to figure out one more side and one more angle. I need to figure out specifically this middle one right here, and I need to figure out this angle over there.
So let's go through it. So our first statement is we give what was first given to us. So what first gave us that L, M is congruent to L, O, and M, N is congruent to O, N. And that was given to us right there.
So we have two sides right now. So we've got to figure out what is that third side right here. So I'm trying to figure out, well, what is this side right here, this line? Well, I see that they both share this, right? So that is ln.
They are both sharing that middle one. So I have ln is congruent to ln. Remember, when the things are equal to each other, that was one of the new properties that we learned. That is called the reflexive property.
So I say the reflexive property. So I think about it, and I have 1, 2, 3. I just found three sides. So those are congruent.
Check mark, I got three sides. So all I've got to do now is find three angles. So again, in the very beginning, it told me that M is congruent to O and MLN is congruent to OLN. And so that was given to me.
So we have two angles. So what we need to find for this third one is we need to find how can... this angle right here be congruent so again remember the new theorem that we just learned that said that in a triangle if two of the sides are congruent then the third side has to be congruent to each other so we have we have angle M and L is congruent to O and L and how do we know that we know that through our new property through our new theorem called the third angle theorem.
Remember that if these two angles are congruent with the other two angles, then that third angle has got to be the same. Remember we did the 30, 50. It has to be 100 for the third one. If these two share the same one, then that last angle has to be the same.
So what does that mean now? That means that we can say that the triangle ln n is congruent to triangle LON. And how do we know that?
We know that through, that's the definition of congruent triangles. So we proved that. We proved all six points.
We proved that the three sides are the same. And we just proved that three angles are the same. So check mark, proved it, got it in the books.
All right, thanks for watching.