WEBVTT mathematics/geometry/pyo
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Welcome back to Educator.com.
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This next lesson is on right triangles; we are actually going to prove right triangles congruent.
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Just like we did a few sections ago, where we used SAS, ASA, AAS, and SSS theorems to prove that two triangles are congruent,
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these theorems and postulates are going to be used specifically just for right triangles.
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The first one is LL theorem: LL theorem is actually Leg-Leg.
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Leg-Leg theorem is when the two legs of one triangle are congruent to two legs of another triangle.
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Again, those other theorems and postulates--SAS, ASA...let me just write them down...AAS, and SSS...these were used
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to prove that two triangles were congruent--just two non-right triangles.
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These work the same way as these, but these theorems are just only for right triangles, proving right triangles congruent.
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These are just to prove triangles congruent.
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So again, the Leg-Leg theorem: If the legs of one triangle (that is AB and BC) are congruent
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to the two legs of the other triangle (and of course, they have to be corresponding: DE and EF), then the triangles are congruent.
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If I mark it like this, if BC is congruent to EF, and AB is congruent to DE, then these triangles are congruent by the LL theorem.
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Now, even though we use the LL theorem because they are right triangles, if you look at this, this is the same thing
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as side, and then angle (because we know that right angles are all congruent, so then angle), side.
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So, even though this is LL, this can also be considered SAS; we just call it LL because they are right angles.
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So again, this is the same as the SAS theorem, but again, it is just because they are right triangles; we just call them LL.
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The LL is the first one; and the next one is HA...let me just write LL so that we keep track of the different theorems
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that we are going to go over, and that we went over...the next one is Hypotenuse-Angle.
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Now, we know that the hypotenuse is the side of the right triangle opposite the right angle.
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This is the right angle; we know that these two are called the legs; and the other side, the long side, the side opposite the right angle, is the hypotenuse.
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So, for the HA theorem, if you want to prove that two right triangles are congruent by the HA theorem,
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we have to prove that the hypotenuses, which are AC and DF, are congruent, and an angle.
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Now, an angle, of course, has to be an angle that is not the right angle.
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So, it is either A and D, because they are corresponding, or angles C and F--it doesn't matter which one.
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I can say that those two angles work; as long as we know that the hypotenuse and an angle are congruent, then these two right triangles are congruent.
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Now, just like the LL, the HA is the same thing as...we have an angle, the next angle, and the next side.
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So, this is the same thing as Angle-Angle-Side; but again, because these are right triangles, you use HA.
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Now, you don't have to use HA, but the reason why we use HA instead of AAS for right triangles is because this is less work.
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In this case, you only have to prove that these two are congruent, in order to prove that these two triangles are congruent,
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whereas, with AAS, you have to prove all three; so it is less work--that is why HA would be the one that you would use for right triangles,
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because you don't have to prove that these right angles are congruent.
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That eliminates one of the angles; so again, it is the HA theorem.
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LL, HA...that was the first one; that was the second one; the next one was the LA theorem.
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L is for leg; A is for angle; again, this is only for right triangles, to prove that they are congruent.
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A leg can be like this one or this; it could be these two legs or these two legs, as long as they are corresponding.
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And there is an acute angle like this, and then an angle...so then LA would be the same thing as Angle-Side-Angle.
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Now, it can be whichever angle; let's say I choose the other angle, or I only have the other angle to work with.
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Then, this is also the same thing as Side-Angle-Angle, or AAS.
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This LA can be the same thing as ASA or AAS; that is LA, a leg and an angle.
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Let's use the Leg-Angle theorem to find the values of x and y.
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Here we have that, let's see...here are the right angles; these two sides are the corresponding sides;
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now, it is not good to assume which sides are corresponding.
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Usually, there will be some kind of indicator to tell you which sides and which angles are corresponding.
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But for this problem, I will tell you that this leg is corresponding to this leg.
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Since they are corresponding, I want to use the LA theorem so that the triangles are congruent.
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Find x and y: that means this 3y - 4 would be congruent to 20, so then you just make that 3y - 4 equal to 20.
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3y...I am going to add 4...equals 24; so y equals 8.
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And then, for x, again (let me just mark these, and then those angles), 2x + 30, that angle, is congruent to 62, that angle, so it has the same measure.
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2x + 30 is equal to 62; so to solve it, subtract 30; 2x = 32, and then x = 16 (divide the 2).
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y is 8, and x is 16; and if x is this value, and y is that value, then these sides will be congruent, and the angles will be congruent;
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and therefore, that is using the Leg-Angle theorem.
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So then, we have another one; first we did HA, then LA and LL, those three; and now we have HL.
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HL, HA, LA, LL: two of them start with the hypotenuse, and two of them start with the leg.
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If the hypotenuse and a leg (it could be either this leg or this leg; it doesn't matter)--any leg,
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so let's just say that is the leg--are congruent, then the triangles are congruent.
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Now, this postulate is the only postulate that is different than a few sections ago when we used SAS, ASA, and all of those theorems.
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This is the only theorem that is different; all of the rest of the other ones were the same.
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They were the same theorems; it is just that they eliminate one of the angles so that it is not as much work,
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because the right angles, we know, are congruent; so then you don't have to prove that part.
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So, they just gave new names for them; this one, however, HL, is the only one that does not apply to both regular triangles and right triangles.
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If you look at this, this is the same thing as Side-Side-Angle.
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Now, if you read this backwards, it is a bad word; so we know, as I mentioned before,
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that if it spells a bad word, it is not one of the ways you prove triangles congruent; it doesn't exist.
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Now, in that case, it would be SSA; but because they are right triangles, this is a special case where we can use HL.
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So then, this one is a little bit different.
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If you are going to use the other theorems and not these, make sure that this one, you know to use for right triangles.
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Again, one pair of corresponding legs, and the hypotenuse, is HL.
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We have HA, HL, LA, and LL.
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For the first example, we are going to state the additional information we need to prove each by the given theorem or postulate.
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We want to prove that these two triangles are congruent by one of the theorems or the postulate.
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But we are missing information; so we have to state what additional information we need.
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Now, before, we went over four--the theorems and the postulate--and they are LL, LA, HL, and HA.
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Think of them as...there are four of them; we can remember that two of them start with an L, a leg; and two of them start with an H.
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They never start with an angle: LL, HA...and there is always a leg or an angle; leg, angle, leg, angle.
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You can only have legs and angles if it is not the hypotenuse: Leg-Leg, Leg-Angle, Hypotenuse-Leg, Hypotenuse-Angle.
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The first one is the LA theorem; we want to prove that these two triangles are congruent by the LA theorem.
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Right now, all we have is that they are both right triangles (obviously--that is the only way that we can use these theorems and postulate)...
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We know that these angles are congruent, because they are vertical angles.
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These are vertical angles, so we have the A; now we need an L.
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So then, the additional information that we need would be the leg.
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It could be either this leg with this leg or this leg with this leg; it doesn't matter.
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In the next one, the HL Postulate is what they want us to use to prove that these two triangles are congruent.
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Now, so far, all we have is this segment right there, because it is the same; they are sharing the same hypotenuse.
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So, that is the hypotenuse for this one, and that is the hypotenuse for this one.
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So, the H is covered; and on a proof, you would say that that is congruent by the reflexive property,
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because the reflexive property says that anything equals itself; so that is the H.
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And then, we are missing the L; we are missing the leg, so either this leg has to be congruent to this leg, or this leg has to be congruent to this leg.
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And then, we can prove that those two triangles are congruent by the HL Postulate.
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And again, this is the only postulate; this is a theorem, theorem...HL, the one that is different,
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the one that is not like SAS, ASA, and all of the others, is a postulate.
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So again, we are missing the leg; that is what we are missing.
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The next example: Find the value of x so that each pair of triangles is congruent.
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For this one, let's see what we are working with.
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We have triangle ABC; we have triangle FED; BC is congruent to DE, because it says 10, and it says 10 here.
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Now, notice how these two triangles don't look exactly the same, but make sure that you don't base it on what it looks like.
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You base it on the facts; so even if you think, "Oh, this side looks like that side," don't assume that they are corresponding or they are congruent.
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You have to look at the numbers; look at what is given to you.
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So, in this case, it is given that BC is 10, and then DE is 10.
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Even if they don't look like they are the same size, just by them having the same measure, they are congruent.
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And so, then we know that we are going to assume that the triangles are congruent, because that way, we can find x.
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So, if these two are congruent, that means that AB has to be congruent,
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because if these two triangles are congruent, then we know that all of the corresponding parts are congruent.
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And that is CPCTC, "Corresponding Parts of Congruent Triangles are Congruent."
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So, we can say that AB is congruent to EF; then, to solve it, I just make 25 - 2x equal to 21.
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I am going to subtract the 25; -2x = -4; divide the -2; x is 2.
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And the next one: we have that the measure of angle A is 60; the measure of angle F is 60;
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so then, those angles are corresponding/congruent.
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And then, the hypotenuse: I make 3x - 7 equal to 2x + 4.
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I am going to subtract the 2x here; that is going to give me 1x.
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I am going to add the 7 here; and then, I am going to get 11; so x = 11.
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OK, we are going to do a two-column proof: this one says that triangle ABC and triangle CDE are both right triangles.
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AB is congruent to DE; whatever is given--just mark it on your diagram; that way, it is easier to see what you are working with.
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And we want to prove that the two triangles are congruent.
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Now, since they are both right triangles, we are going to use one of the four theorems and postulate that we just went over.
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If they were non-right triangles, then we would use SAS, ASA, AAS, SSS...
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The four that we went over are LL, LA, HL, and HA; here is the postulate.
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We need one more corresponding part to be congruent; we have a leg; that means we are going to use either this one or this one.
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We need one more thing, and that is all of the information that they give me.
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So then, I have to think, "OK, what other information do I have--what can I tell from this diagram?"
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And I know that vertical angles are congruent; that means that here, even if they don't tell me that they are congruent,
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I can assume that they are congruent, because they are vertical, and we know that vertical angles are congruent.
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Now, we have two corresponding parts that are congruent; and that is a leg and an angle.
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We are going to use this one: #1: here are the statements and the reasons.
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#1: Triangle ABC and triangle CDE are right triangles, and then that AB is congruent (I don't know why these won't work) DE.
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And that is all "given."
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The next one: Here is my L, leg; then, angle ACB is congruent to...and if I am going to say ACB,
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then remember: the other angle that I name has to be corresponding, so then I am going to say angle DCE,
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because D and A are corresponding; they are right angles--angle DCE.
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Again, I didn't have to say angle ACB; I could have said angle BCA.
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If I say angle BCA here, then for the next one, I have to say angle ECB; whatever is corresponding with B, I have to state in that same order.
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Then, that would be the reason "vertical angles are congruent."
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Here is my A; I have both L and A; that means I can go ahead and say that the triangles are congruent.
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Triangle ABC is congruent to triangle CDE, and the reason for that is "LA theorem."
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The next example is also a proof: let's see what we have.
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The given is that this angle and this angle are congruent; they want us to prove that AD is congruent to CB, that these two legs are congruent.
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Now, there is no way to just prove directly that those two legs are congruent.
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So then, I have to say, "OK, can I first prove that the triangles are congruent, and then say that corresponding parts of congruent triangles are congruent?"
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If you are given two triangles, and then they want you to prove that a pair of corresponding parts are congruent,
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then if you can't see a way to do it directly, then you have the other option
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of first proving the triangles congruent, and then saying that corresponding parts of congruent triangles are congruent.
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That is what we are going to do here: so then, we have to think, "How do we prove the triangles congruent?"
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Well, if the two triangles share a side, then that side is automatically congruent, because of the reflexive property.
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So again, my four are LA, LL, HA, and HL.
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Now, this is going to be a leg of this triangle and a leg of that triangle.
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So, I have a leg (that means that I am going to use one of these two) and an angle; so I am going use the LA theorem again.
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And then, once I have proved that those two triangles are congruent by the LA theorem,
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then I can say that AD is congruent to CB, because they are corresponding parts.
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And if you have congruent triangles, then all corresponding parts are congruent.
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Statements, and reasons on this side: Angle A is congruent to angle C; that is my angle, and then "given."
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The next one: BD is congruent to...this one is a little bit...the way you say it...BD is actually going to be congruent to DB.
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BD is going to be congruent to DB; and the reason why it is not BD to BD:
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if you flip these around, then the B in this triangle is actually corresponding to the D in this triangle.
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So, if you are going to say BD, then you have to say DB.
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And then, the reason, again, is the reflexive property, and then that is your leg.
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So then, my triangles are now congruent: triangle ABD is congruent to triangle...corresponding to A is C; with B is D; and then B again.
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So then, again, B and D are corresponding; B of one triangle is congruent to D of the other triangle.
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And then, that would be the LA theorem.
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And then lastly, remember: we have to start here as our first step, and then we have to end here as our last step.
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Then, our last step is going to be that AD is congruent to CB; the reason for that is CPCTC.
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Again, the four theorems that we are going to use are LA, LL, HA, and...the last one is actually not a theorem; HL is a postulate.
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So again, these theorems and postulate can be used to prove that two right triangles are congruent.
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You could use the other theorems, but that is just going to be more work,
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because then you have to prove that an additional angle would be congruent,
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whereas with these, you only have two, because the angle is already covered, because they are right.
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We have LA, LL, HA, and the HL postulate.
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That is it for this lesson; we will see you for the next lesson, too--thank you.