WEBVTT mathematics/geometry/pyo
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Welcome back to Educator.com.
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The next lesson, we are going to explore congruent triangles.
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We are going to be looking at triangle parts, and then we are going to compare two triangles together
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to see if we can see that they are congruent.
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In order for triangles to be congruent, they have to have the same size and shape,
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because if two triangles just have the same shape, then one can be this small, and another one can be this big.
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They have the same shape, but they are not the same size.
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They have to be the same size and shape.
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Here we have two triangles, triangle ABC and (it is congruent to) triangle DEF.
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Now, if you want to state triangles, remember: you can just write a little triangle symbol in front of that.
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And so, that is how you write "triangle ABC" in symbols.
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Triangle ABC is congruent to triangle DEF; these two triangles are congruent.
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Now, in order for two triangles to be congruent, they don't have to just look and be in the same upright position.
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Here there are three different ways that triangle DEF is shown.
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And even if you move it around, and you flip it, and you do all of these things to it,
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which are called congruence transformations, you are still going to get a congruent triangle.
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It is still going to be the same thing; so from this triangle, DEF, to this triangle, this is when it just moves.
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And when it just moves to a different position, that is called **sliding**; so this is sliding.
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Nothing changes about it, but it just moves.
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The next one, right here: see how it looks like it is turned a little bit.
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That is kind of doing one of these; and that is **rotating**; this one is rotated.
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And that is just when you take the triangle and just move it so that this top angle is no longer the top angle; you are just rotating it.
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And the next one, right here: this one is **flipping**; all we did was to take this, and we just flipped it from up to down.
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You just flip it; so this one is kind of like that--you are flipping it.
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There are three congruence transformations: slide, rotate, and flip.
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And no matter what you do to it, it is always going to be congruent to triangle ABC.
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This triangle, DEF, is still congruent to this triangle, ABC.
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So that means that, if I have triangle ABC like this, and triangle DEF like this, they are still congruent,
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even though they are not in the same upright position, like these two.
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These are congruence transformations.
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Now, **corresponding parts** of triangles have to do with their angles; you are comparing the angles of one triangle
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to the angles of another triangle, or the sides of one triangle to the sides of the other triangle.
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So, the parts of a triangle, we know, are angles and sides; and you are saying that,
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when it is corresponding...remember corresponding angles?--if we have two parallel lines and a transversal,
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remember how the corresponding angles were the angles in the same position?
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If we had an angle on the top right from the top part, or just one of those, then the bottom for the next part...
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let me just draw it out for you: remember: this angle right here (if these lines are parallel) and this angle were corresponding angles.
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This angle is the top right, and this angle is also the top right; so they have the same position.
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It kind of means the same thing here, too.
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Again, we have that triangle ABC is congruent to triangle DEF.
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Now, remember: we have six parts total: three angles and three sides--six parts total.
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That means that, if I have two congruent triangles, this triangle congruent to this triangle, then all of its parts...
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see how, for this triangle, A is named first; it doesn't have to be named first, but it is.
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Then, this D is named first for this triangle; then angle A is congruent to angle D.
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This angle is corresponding to angle D.
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So, to write "corresponding," I can draw an arrow like that, and that means "corresponding"; "corresponding" is like that.
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Now, B is second, and E is second; that means automatically that this angle and this angle right here are corresponding.
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B is corresponding to E; and then, the last angle that is mentioned, C, is corresponding to angle F; C is corresponding to F.
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That means that angle C is congruent to angle F.
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So then, this just shows that they are corresponding; I can also write the same thing, showing that they are congruent.
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I know that, because we know that the triangles are congruent, all of their parts are congruent.
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So, angle A is congruent to angle D; angle B is congruent to angle E; and angle C is congruent to angle F.
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Now, each one of these letters represents an angle.
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And if you put two of them together, they represent sides.
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Whatever is written first is going to be congruent to this one that is written first.
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So then, I can't say that angle A is going to be congruent to angle F, because they have to be named in the same order.
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Congruent parts have to be named by that same order.
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Now, if I were to say triangle BCA, that is OK; I can name this triangle however I want.
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I can name it triangle ABC; I can name it triangle BCA; I can name it triangle BAC, triangle CAB, CBA...whatever.
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I can name this triangle however I want, but if I have two congruent triangles,
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and I am going to write a congruence statement, then whatever I write next is going to depend on what I wrote first.
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If I am going to write it like this, then I have to write what is congruent to angle B (angle E) first; C is F; A is D.
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You can write it like this, or you can write it like this.
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If you decide to name this triangle CAB, then you would have to say that it is congruent to triangle FDE.
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So again, it doesn't matter how you name the first triangle; but the second triangle will be dependent on how you write that first triangle.
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So, again, here are the corresponding angles; and then, for the sides, AB is congruent to DE.
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And you can also look at that in this way, too: AB is congruent to DE; BC is congruent to EF;
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now, if I said BC, then I can't say FE, because again, remember: B is congruent to E;
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so if you are going to say BC, then the same thing applies for this.
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You can't say FE; you have to say EF, because B and E are corresponding.
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It has to be in the same order.
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And then, the next one: AC is congruent to DF; if you said CA, then you would have to say FD,
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because C and F are congruent; so CA would be congruent to FD, and so on.
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So then, those are my six congruence statements for my corresponding parts.
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Here is the congruence statement for the congruent triangles.
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And then, here is the symbol to write that the angles are corresponding, just to show correspondence.
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So then, these are corresponding parts; again, you have to make sure that you name them in the order of its congruence.
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If you don't have a diagram--let's say you only have this congruence statement,
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and you don't have a diagram, but you have to name all of its congruent parts;
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then your angles, remember: the first angle, angle A, is going to be congruent to this angle, D;
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angle B is congruent to angle E; angle C is congruent to angle F.
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But then, for the sides, if I say side AC, then that would be congruent to the side DF.
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So it is first/third, then first/third on this one.
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If I said CB (that is third/second), then it has to be third/second on this one, FE.
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So, that is how you can just name the parts of it.
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Definition of congruent triangles: Two triangles are congruent if and only if their corresponding parts are congruent.
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So, here I have two triangles: this can go both ways--"if and only if" means that you could have two conditionals;
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if two triangles are congruent, then their corresponding parts are congruent; that is one way;
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and then the converse would be that, if the corresponding parts are congruent, then the two triangles are congruent.
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So, "if and only if" just means that this statement, as a conditional, is true, and its converse is also true.
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When you switch the "if" and "then," that is also true; that is "if and only if."
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So, if all of these corresponding parts, meaning all six parts of the corresponding parts
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from the one triangle to the other triangle, are congruent, then the two triangles are congruent.
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If you say that the two triangles are congruent, then all of its corresponding parts, all of the six parts, will be congruent.
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So, if triangle GHL is congruent to this (and again, I have to write it so that it is corresponding with this:
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G with M, H with P, and L with Q), then all of its corresponding parts are congruent.
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Or you could say, "If all of the corresponding parts are congruent," if angle G is congruent to angle M,
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and angle H is congruent to angle P, and so on and so on; then this triangle is congruent to this triangle.
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And that is by the definition of congruent triangles.
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Now, another name for this is CPCTC; this is very, very important for you understand in geometry.
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CPCTC means Corresponding Parts of Congruent Triangles are Congruent.
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Try to say that a few times, just so that it will sound familiar.
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Corresponding parts of congruent triangles are congruent; so this is saying this right here, what we just said.
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If the triangles are congruent, if somewhere it is stated--if it is given to you, or you proved it,
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or whatever way you figured out that those two triangles are congruent--from there, then all of the corresponding parts are congruent.
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They are congruent to each other--the corresponding parts of each of these triangles.
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Again, CPCTC says that, if those two triangles are congruent, then corresponding parts are congruent.
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Corresponding parts of congruent triangles--"of congruent triangles" means that it has to be congruent first.
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Triangles have to be congruent first, and then all of the corresponding parts are congruent.
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If I wanted to tell you that these two triangles are congruent, and then I said, "OK, well, angle L is congruent to angle Q,"
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what proof do I have--what is my reason behind angle L being congruent to angle Q?
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Well, I can just say "CPCTC," because since these triangles are congruent, then all of their corresponding parts are congruent.
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So, once it is stated that those two triangles are congruent, from there I can say that any of its parts are congruent.
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And then, my reason will be "CPCTC."
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Again, if the triangles are congruent, then all of their corresponding parts are congruent.
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From there, you can say that any part...I can say that HG is congruent to PM, or HL is congruent to PQ, and so on.
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And the reason why I am emphasizing this is because this is something that students make mistakes on all the time.
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Using CPCTC is a little bit hard to understand, especially when you have to use it for proofs.
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Just remember: with CPCTC, triangles have to be congruent first;
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and once they are congruent, then you can say that any of their corresponding parts are congruent.
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OK, triangle congruence: now, we know the reflexive property, symmetric property, and transitive property.
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Now, this is a theorem, so this is actually supposed to be proved.
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But I am just going to explain to you that we can now apply the reflexive property, symmetric property, and transitive property to congruent triangles.
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You know how the reflexive property is when you have triangle ABC congruent to triangle ABC.
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And that is the reflexive property; the symmetric property is that, if triangle ABC is congruent to triangle DEF,
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then triangle DEF is congruent to triangle ABC.
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And then, the transitive property: if triangle ABC is congruent to triangle DEF, and triangle DEF is congruent
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to triangle GHI, then triangle ABC is congruent to triangle GHI.
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So, you can apply the reflexive property, symmetric property, and transitive property to congruent triangles.
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OK, let's go over our examples: Write a congruence statement for each of the corresponding parts of the two triangles.
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"For each of the corresponding parts" means that we have six total.
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Now, I don't have diagrams, so I can't see which sides are congruent to the other sides,
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and so on; so I have to just base it all on this congruence statement right here.
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I know that angle D is congruent to angle X; angle E is congruent to angle Y; and angle F is congruent to angle Z.
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There are three parts; then, DE is congruent to XY; how do I know that?--because DE is here, and XY is there;
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and then, let's see: EF is congruent to...EF is this, so YZ; and DF (that is DF right there--
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don't think that that is the symbol for a segment) is congruent to (that is D and F right there, so it has to be) X and Z.
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Now, if it is DF, can I say ZX? No, if I am going to say D first, then it has to be "DF is congruent to XZ."
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Now, if this seems like it is really picky, and it is just too much...it is not too bad.
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It is just that you have to write this congruence statement, because you are stating that they are congruent.
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And if you are going to say that they are congruent, then you have to make sure that you are stating that the right parts are congruent.
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It is something that is just necessary.
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DF is congruent to XZ; and that is it--we have all six corresponding parts.
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Let's say that I want to just do one more thing: if I want to rewrite this congruence statement to how I like it--
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let's say that I don't like that it is in the order DEF--then I can just...let's say I want to do triangle FDE;
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I can rewrite this triangle congruence statement the way I want it, as long as I change the second part, too.
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I can't keep it XYZ: so if I say FDE, then I have to write...what is corresponding with F? Z; triangle Z...what is corresponding to D? X; with E, Y.
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So, I can write it like that, too.
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The next example: Write a congruence statement for the two triangles.
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I am going to say that this is congruent to this, is congruent to this; this is congruent to this.
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And these slash marks: if I have one, all of the segments with one slash mark are congruent;
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all of the segments with two slash marks are congruent; and then, all of the segments with three are going to be congruent.
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Let's say the same thing with this: if I have one mark there, all of the ones with one are congruent;
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all of the ones with the two marks are congruent; and then, all of the ones with the three are congruent.
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So now, I have all six parts of this triangle being congruent to the corresponding parts of the other triangle.
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Now, I can write a congruence statement: Triangle SPT is congruent to triangle...what is corresponding with S? R.
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OK, R is congruent to S; what is corresponding with P?--M; and then, with T is T.
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Now, if you did this same problem on your own, then your congruence statement might be different than mine.
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It is OK, as long as you make sure that these three parts are corresponding to these three parts.
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If you have triangle PTS, then it has to be triangle MTR.
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Draw and label a figure showing the congruent triangles.
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I can just draw triangle...I can just do like that...here is MPO, TSR, and then I want to label and show the congruent triangles.
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MP is congruent to TS; and then, PO is congruent to SR; MO is congruent to TR.
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And then, M is congruent to angle T; P is congruent to angle S; and O is congruent to angle R.
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That would be showing a diagram of this congruence statement.
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Draw two triangles with equal perimeters that are congruent.
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If I said that this is 4, 5, 6; then this also has to be 4, 5, 6, because they are going to be congruent.
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And then, the corresponding angles would be congruent.
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So, in this case, they are going to have the same perimeter, because with perimeter,
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remember: we add up all of the sides, so 4 + 5 + 6 is going to be the same as 4 + 5 + 6.
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So, this would be an example of two triangles with equal perimeters that are congruent,
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because I showed that all three corresponding sides are congruent, and the corresponding angles are congruent.
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Draw two triangles that have equal areas, but are not congruent.
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Let's say I have a triangle here, and say that this is 8, and this is 5.
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Then the area here is going to be 20 units squared, because 8 times 5 is 40, divided by 2 is 20.
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So, that is one triangle; and then, another triangle--let's say the same area; that means, since these add up to 40,
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that this also has to add up to 40, the base and the height.
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Let's say this is 10, and this is 4; and that means that the area is...40 divided by 2 is 20 units squared.
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So again, they have the same area, but they are not congruent, because...see how this side and this side are different.
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And then, I can just say that this is, let's say, 7 and 9; this would be 6 and 5.
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That is it for this lesson; thank you for watching Educator.com.