Understanding Triangle Congruence Postulates

In this video, we're going to focus on proving if two triangles are congruent. Now, there's four postulates that you need to know. The SSS postulate, SAS, ASA, and AAS. So the first one tells us that if the two triangles have three sides that are congruent, then the two triangles are congruent, which means every part of those two triangles are congruent. Next, if you have two sides and the included angle, then the two triangles are congruent. A stands for angle, S stands for side. The next one, angle, side, angle. It has to be in order. And then the last one, angle, angle, side. So what I'm going to do is I'm going to give you a few triangles, and I'm going to ask you which postulate applies to those two triangles that can prove that they're congruent. So here's the first one. So let's call this triangle ABC and triangle DEF. So let's say if AB is congruent to DE, BC is congruent to EF, and AC is congruent to DF. So what postulate proves that these two triangles are congruent? Well, we have a side, a side, and another side. So, we can use the SSS postulate to show that those two triangles are congruent. So, we can therefore make the statement that triangle ABC is congruent to triangle DEF. And make sure each letter corresponds to the same thing. So, for example, A corresponds to D. So, S will be the first letter listed. B corresponds to E, so that's the second letter. And the last one, C corresponds to F, which is the third letter. Alright, so now let's look at another example. So consider these two triangles. And let's call it triangle RST and VXY. So let's say that angle S is congruent to angle X, and RS is congruent to VX, and angle R is congruent to angle V. What postulate proves that these two triangles are congruent? So we have an angle, a side, and an angle. You want to trace it in one direction. You don't want to jump around. So therefore, we have the ASA postulate, angle, side, angle. And so we can make the statement that triangle RST is congruent to triangle VXY. So again make sure that the letters match. So R corresponds to V, S corresponds to X, and T corresponds to Y. So let's look at another example. So let's call this triangle DEF. triangle ABC and let's say that DE is congruent to AB and DF is congruent to AC an angle D is congruent to angle a what postulate can we use to prove that these two triangles are congruent So we have a side, an angle, and a side. So it's side, angle, side, S-A-S. And so we could therefore say that triangle D-E-F is congruent to triangle A-B-C. And I forgot the triangle symbol. What about this one? Let's call this triangle XYZ and let's say this is ABC. And let's say that XZ is congruent to AC and angle Z is congruent to angle C. and angle B is congruent to angle Y. What postulate can we use to prove that these two triangles are congruent? So we have an angle, an angle, and a side. So by the AAS postulate, these two are congruent. So triangle XYZ is congruent to triangle ABC. Now sometimes you may have a composite triangle. That is two triangles within a single triangle. So let's say that this is A, B, C, and D. And let's say that we're given that AD is congruent to CD, and AB is congruent to BC. What can we prove? What conclusions can we draw? So let's separate this composite triangle into two triangles. So the first one on the left is triangle ABD. On the right, we have triangle CBD. So we know that AD and DC are the same, AB and BC are the same, but notice that they share a common side, and that is DB is equal to DB, because that's the same segment. So therefore, we have two triangles that share, that have the same three sides. So these two triangles are congruent by the SSS postulate. So sometimes if you don't see the two triangles, you may want to split this composite triangle into two smaller triangles, if that helps. So we can make the statement that triangle ABD is congruent to triangle CBD. Let's try another composite figure. Let's call this A, B, C, D, and E. So let's say that AC is congruent to EC, and also angle A is congruent to angle E. What can we do to prove that the two triangles are congruent? The triangle on the left, ACB, and the triangle on the right, ECB. Right now we only have two things that are congruent. We have an angle and a side. Is there anything else that we could see that's congruent? Now whenever you have two lines that intersect each other, or two segments, they will produce something that has vertical angles vertical angles are congruent so in this picture angle one is congruent to angle 2 and 3 is congruent angle 4 so therefore angle a CB is congruent to ECD and we now could prove that the two triangles are congruent by the ASA postulate So we have two angles and the side between the two angles. So now we can say that triangle ACB is congruent to triangle ECD. Notice that E and A are the same in this particular example. And D and B correspond to each other. So let's say that this is triangle ABC and triangle DEF. Now let's say that angle A is congruent to angle D, and that angle B is congruent to angle E, and angle C is congruent to angle F. So are these two triangles congruent? so we have an angle and angle and an angle so is there such a possible at the AAA possible no there isn't so therefore these two triangles are not congruent so we can write this statement triangle ABC I put S for some reason. Let's correct that. It is not congruent to triangle D, E, F. So let's say this is X. YZ, let's call this A. And in this example, we're going to say that angle X is equal to angle Z, and XYA is congruent to ZYA. And with this information, what can we conclude about these two triangles? Can we say that triangle XYA is congruent to triangle ZYA? Now remember, they share a common side, that is YA. So we could say that YA is congruent to YA for the two small triangles based on a reflexive property. Thus, we have an angle, an angle, and a side. So by the AAS postulate, which is one of the four that we mentioned in the beginning, these two triangles are congruent. So triangle X, Y, A is congruent to triangle Z, Y, A. So let's say this is Q, R, S, T. Let's say X. And let's say that QS is congruent to ST and RS is congruent to SX. Prove that these two triangles are congruent. So as you recall, angle RSQ is a vertical angle with XST. Those two are vertical angles, which means that they're congruent. So we have a side, an angle, and a side. So by the SAS postulate... we can say that triangle RSQ is congruent to triangle XST. So notice that R and X are opposite to the same side. So therefore, R and X are congruent. And T... is opposite to the same side as Q. So Q and T are congruent. So therefore triangle R S Q is congruent to triangle X S T. Now let's look at one last example before we work on two column proofs. Let's call this A, B, C, and D. And let's say that AB is congruent to AD, and BC is congruent to DC. What do we need in order to prove that these two triangles are congruent? Now, they share the same side, side AC. So therefore, we have a side, side, and a side. So by the SSS postulate, we can say that triangle ABC is congruent to triangle ADC. And that's it. Now let's work on some two-column proofs. Let's say this is A, B, C, and D. And we're given that segment AD is congruent to CD. And also that B is the midpoint of AC. Your task is to prove that the two triangles are congruent. That triangle ABD is congruent to triangle CBD. So feel free to pause the video if you want to. So let's write our two columns. Statements. and reasons. So the first thing you should do as always is rewrite what you're given. So we know that AD is congruent to CD, and that's a given. And we know that B is the midpoint Let me just mark that AD is congruent to CD. Now, B is the midpoint of segment AC. So, that's given. Now, what are some conclusions that we can draw if B is the midpoint? If B is the midpoint, that means that AB is congruent to BC. So, let's write that in step 3. So, AB is congruent to BC. And the reason? Definition of a midpoint. Now, let's get rid of this stuff. It's just taking up space. Now, what else can we say? So right now we have two sides. We either need another angle or another side. Now notice that BD is shared between both triangles. So that's the third side. That's the common side. So we could say that BD is congruent to itself. Whenever you have a common side, the reason you can always write is the reflexive property. Now what postulate can we use to prove that these two triangles are congruent? So notice that we have a side, a side, and another side. So we can make the final statement that triangle ABD is congruent to triangle CBD, and the reason for that is the SSS postulate. And... We're using statements 1 that show that these two sides are congruent, and statement 3, which show that AB is congruent to BC, and also statement 4, where BD is congruent to itself. And so we're going to write 1, 3, and 4. And so that's how you can prove that two triangles are congruent. You can use the SSS postulate, SAS, ASA, or AAS. By the way, I recommend that you pause the video for each of these problems. And it's best if you work on it, because that's the way you're going to learn. Pause the video, try the problem, and then once you get the answer, unpause it to see the solution. Now, you might have a different way of getting to the answer, so sometimes there's more than one way. So I just want to let you know about that. Now let's go ahead and begin. So let's call this MOP, and this is going to be R. and we're given that segment RO is perpendicular to segment MP and also segment MO is congruent to segment OP. Your task is to prove that triangle MRO is congruent to triangle PRO. So go ahead and begin. See if you can do this. So let's write our two-column proof. So as always, what's the first thing that we should do? Remember the first thing you should do is write the given. So first let's start with the fact that segment MO is congruent to segment OP and that's given. And just to keep things in perspective just to make it easier, let's highlight that. So here's MO and that's congruent to OP. Let's highlight it with tick marks. Now let's move on to statement 2 and that is segment RO is perpendicular to segment MP. So that's a given. Now this is RO and this is MP. So they're perpendicular. What conclusions can you draw from perpendicular lines? At what angles do perpendicular lines meet? A perpendicular line will always form a 90 degree angle with the line that intersects with. So, the angle between RO and MO is 90. The same is true between RO and OP. Perpendicular lines intersect at right angles. So, therefore, we could say that... angle MOR is congruent to angle POR and the reason we can say definition of perpendicular lines or simply that perpendicular lines form right angles Now what else can we say? Now notice that we do have a common side. So for statement 4 we can say that RO is congruent to RO. And we know the reason for that. That is the reflexive property. So now, we have a side, we have an angle, and another side. So we can make the statement that triangle MRO is congruent to triangle PRO. And this is based on the SAS side angle side postulate. And so we've used, in this example, statement 1, which showed that MO is congruent to OP. We use number 3. the two angles are congruent, and number 4 for the common sign. So this is based on statements 1, 3, and 4. And so that's it. Now let's move on to our next example. So let's say this is A, B, C, D, and E. We're going to call this angle 1, angle 2, 3, and 4. So given that angle 1 is congruent to angle 4, and also that segment AC is congruent to segment EC, prove, go ahead and prove that Triangle ABC is congruent to triangle ECD. So once again, pause the video and go ahead and work on this problem. So hopefully by now, you're getting the hang of how to do these kinds of problems. So I'm going to give you a lot of practice problems so you can master this topic. So, statement 1, we can say that angle 1 is congruent to angle 4. And that statement was given to us. So now let's mark it on the diagram. So, angle 1 is congruent to angle 4. now statement to we know that segment AC is congruent to segment EC and that's given to us as well so here is segment music different color this is segment AC and here is easy now what else can we say about the two triangles Now notice that we have vertical angles. Angle 2 is congruent to angle 3. And why can we say that? Why is angle 2 congruent to angle 3? These two are vertical angles. Remember vertical angles are congruent. So now we have three things. We have an angle, we have a side, I mean a side by the yellow mark there, and we have another angle. So by the ASA postulate, we can say that triangle ABC is congruent. It's a triangle E, C, D. And the numbers that we use are basically 1, 2, and 3. It's based on statements 1, 2, 3. Alright, let's try another example. So let's say this is A, B, C, D, and E. So we're given that segment AB is congruent to CD. And segment AE is congruent... to segment DE. And also that angle 1 is congruent to angle 4. And I need to write those angles. This is angle 1, 2, 3, and 4. And 3 is on the inside. Your task is to prove that triangle ABE is congruent to triangle DCE. So go ahead and take a minute and work on that example. So how should we begin? Well, let's write the two columns, statements and reasons. So the first statement we're going to say is that AB is congruent to CD, which is given. And as always, we're going to mark this on the graph. So this is AB, and here's CD. Now the second statement is that AE is congruent to DE, which is also given. And so this is AE, and this is DE. Now we're given that angle 1 is congruent to angle 4. So we're going to write that as well. Angle 1 is congruent to angle 4. So these two are the same. Now, if those two angles are the same, then the angles that are supplementary to those angles 2 and 3 are the same as well. So in step 4, we can say that angle 2 is congruent to angle 3. And the reason for that is that supplements of congruent angles are congruent. So let's say if angle 1 was 120, angle 4 has to be 120. That means number 2 has to be 60. 60 plus 120 is 180. So 3 is 60, which means 2 equals 3. Now, step 5. We can make the final statement that triangle ABE is congruent to triangle DCE. Now, what postulate can we use to show that those two triangles are congruent? So, let's see. We have, let's mark that these two are congruent, angles 2 and 3. So we have a side, an angle, and a side. So hopefully you see it. You can also see on this side too. Side, angle, side. So it's the SAS postulate. And we've used statements 1, 2, and 4 to accomplish the congruence of these two triangles. So it's 1, 2, and 4. And that's it for this problem. Here's another one that we can work on. Similar to the first two, but with a few differences. So let's call this T, Q, R, and S. And we're given that angle T, Q, R. is congruent to angle TSR. And we're also given that TR bisects angle QTS. So your task is to prove that triangle QTR is congruent to triangle STR. So go ahead and work on it. So, number one. The given, TQR is congruent to TSR. So TQR, we could describe it as angle Q. And TSR, we could say is just angle S. Because angle Q can only correspond to this angle. So I'm going to say angle Q is congruent to angle S. Let's write it. And the reason is that it's given. now number two we know that ray TR bisects angle QTS so this is QTX and this is ray TR it bisects it or splits it into two congruent angles which means that these two angles are congruent Let's put the reason for statement 2 as given. In statement 3, we can say that angle QTR is congruent to angle STR. Now, the reason for this that we can write is based on the definition of an angle bisector. This is an angle bisector. Angle bisectors are rays, and they split an angle into two congruent angles. Now, we do have a common side. So every time you have a common side, you can use the reflexive property. So we're going to state that TR is congruent to itself, TR. And that's the reflexive property. Now finally, we can make the fifth statement that triangle QTR is congruent to triangle STR. And what postulate can we use to say this? So here we have... an angle, another angle, and a side. So this is the AAS postulate. And we use number 1 to prove that angle Q is equal to angle S, and number 2, that just helps us to establish number 3 which... We did use that. That's the second angle. And for the side, that's number 4. So it's based on statements 1, 3, and 4. And that's how you can prove the two triangles, using the AAS postulate, angle-angle-side. Now let's move on to our next problem. So let's call this A, B, C, D, and E. And you're given that C is the midpoint of two segments. C is the midpoint of A, E. and it's the midpoint of BD. So with this information, go ahead and prove that triangles ABC is congruent to triangle EDC. So how can we prove that triangle ABC is congruent to triangle EDC? What would you say? So let's start with the given. So we're given that C... is the midpoint of AE and BD. So that's a given. Now, number two, because C is the midpoint of AE, that means that AE is bisected into two congruent segments. So we could say that AC is congruent to EC. And the reason is definition of a midpoint. Now, number three, we could say that BC is congruent to CD. Because C is still the midpoint of BD. And the reason for that is the same. Definition of a midpoint. Now what else can we say? Another statement that we can make... is that angle ACB is congruent to angle ECD. Now, I want to mention that A corresponds to E, not D in this case. Because look at the angle, or rather, the side that angle A is across. Angle A is across the side with two marks, and angle E is across the side with two marks. So if these two sides are the same, the angles that are opposite to them should be the same. So that's how we know to match A with E and B with D. So those two angles are congruent because the vertical angles are congruent. So now we have enough information to prove that triangle ABC is congruent to triangle EDC. So we have a side, an angle, and a side. So using the SAS postulate. So we've used statement 2, 3, and 4 to prove that these two triangles are true. So let's write that here. And that's it for this problem. So let's say that this is ABC and this is going to be DEF and we're given that AD is congruent to CF and also that CB is congruent to DE. and AB is congruent to EF. Your task is to prove that the two triangles are congruent. So we're going to try to prove that triangle ABC is congruent to triangle FED. So let's start with the fact that AD is congruent to CF and that's given. And number 2, CB is congruent to DE. So that's given as well. And 3, AB is congruent to EF. Now the reason why I wanted to write all of that at the beginning is to make some more space. So we know that AD is congruent to CF. Now it might be helpful to draw two separate triangles. So this is going to be triangle ABC and that's triangle FED. Let me use a different color for that. Now, based on statement 2, CB is congruent to DE. So here's CB and here's DE. Those two sides are congruent. And AB is congruent to EF. So this is AB and this is EF. So if we can prove that AC is congruent to DF, then we can prove that the two triangles are congruent using the SSS postulate. So how can we prove that these two segments are congruent? What can we do? So notice that we have an overlapping segment, DC. So that implies we need to use segment addition of some sort. First, we're going to say that AD is equal to CF before we perform some mathematical operation. We can say this based on definition of congruent segments. To get AC and DF, what I need to do is I need to start with this sentence or this equation and add segment DC to both sides. I'm going to have AD plus DC, and that's equal to CF plus DC. Now, because I added it to both sides, this is simply the addition property. Now, notice that AD plus DC is AC. So, I can replace the left side with AC. And DC plus C... is DF so therefore I can replace the right side with DF and whenever you add in two small segments to get a larger segment this is known as segment addition now step 7 I can say that AC is congruent to DF and this is definition of congruent segments so now I'm going to put the mark so this is AC and this is DF so now we can make the final statement that triangle ABC is congruent to triangle F ED And the reason for that is the side, side, side postulate. And so we've used, let's see, number 2, CD is equal to, CB is equal to DE. And we've used number 3, AB is equal to EF. And also number 7. So it's 2, 3, and 7. And that's it for this problem. Now this is going to be the last problem of the video. So this is going to be A, B, C, D, and E. And so here's what we're given. We're given that B and C, that is points B and C, trisect segment AD and then we're given that angle A is congruent to angle D and also angle EBC is congruent to angle ECB so go ahead and prove that triangle EAB is congruent to triangle EDC. So take a minute and work on this problem. So let's start with the first one. Let's say that angle A is congruent. to angle D and that is given. And let's mark it on the graph. So here's A and here's D. They're congruent to each other. Now number two, we know that angle EBC is congruent to ECB. So this is EBC. So that's congruent to ECB. Let me just use a different color. Instead of having so many marks. So if those two angles are equivalent, what can we conclude? Remember, our goal is to prove that triangle EAB is congruent to EDC. So we don't need these two angles. It's outside of the triangles that we want. However, we do want the angles that are supplementary to those angles. So that means that angle EBA is congruent to ECD. because there's supplements of congruent angles. So let's write that. Statement 3. Angle EBA is congruent to angle ECD. And the reason? Supplements of congruent angles are congruent. Now, we just need one more piece of information, and it has to do with this. So, let's rewrite that statement. So, number 4, points B and C trisect AD. So, that's given. And what is the conclusion of that? Well, when something bisects a segment, it splits it into two equal parts. When you have two points that trisect the segment, it splits it into three equivalent parts. So, AB and BC and CD are equal to each other. Bar focus, well we can write all of it. So AB is congruent to BC, which is congruent to CD. And that's the definition of an angle trisector. Or rather, not an angle trisector, but a segment trisector. Because we're trisecting a segment, not an angle. So now we have everything we need. We have an angle, a side, and an angle. And the same is true for the other triangle. Angle, side, angle. So now we can make the statement that triangle EAB is congruent to triangle EDC and this is based on the angle side angle postulate and we've used statement one that is a equals D and we also use a statement 3, EBA is equal to ECD and also statement 5 where AB is congruent to CD. So this is 1, 3, and 5. And so now that is it. So hopefully this video gave you enough practice where you can master the ability of proving if two triangles are congruent. So the next section is C. Cp, Ct, C. Corresponding parts of congruent triangles are congruent. which is based on this video. So basically, everything you've learned in this video, you're going to need it for the next video, or for the next topic. So once you prove that two triangles are congruent, you can prove that any other part is congruent. For example, in the next section, you just got to take it a step further. I could say that AE is congruent to ED. And all I gotta say is C-P-C-T-C. If two triangles are congruent, then the corresponding parts are congruent. So that's the next topic though, which you can look for my video on YouTube. Thanks for watching.

In this video, we're going to focus on proving if two triangles are congruent. Now, there's four postulates that you need to know. The SSS postulate, SAS, ASA, and AAS.

So the first one tells us that if the two triangles have three sides that are congruent, then the two triangles are congruent, which means every part of those two triangles are congruent. Next, if you have two sides and the included angle, then the two triangles are congruent. A stands for angle, S stands for side.

The next one, angle, side, angle. It has to be in order. And then the last one, angle, angle, side.

So what I'm going to do is I'm going to give you a few triangles, and I'm going to ask you which postulate applies to those two triangles that can prove that they're congruent. So here's the first one. So let's call this triangle ABC and triangle DEF.

So let's say if AB is congruent to DE, BC is congruent to EF, and AC is congruent to DF. So what postulate proves that these two triangles are congruent? Well, we have a side, a side, and another side. So, we can use the SSS postulate to show that those two triangles are congruent.

So, we can therefore make the statement that triangle ABC is congruent to triangle DEF. And make sure each letter corresponds to the same thing. So, for example, A corresponds to D. So, S will be the first letter listed. B corresponds to E, so that's the second letter.

And the last one, C corresponds to F, which is the third letter. Alright, so now let's look at another example. So consider these two triangles. And let's call it triangle RST and VXY. So let's say that angle S is congruent to angle X, and RS is congruent to VX, and angle R is congruent to angle V. What postulate proves that these two triangles are congruent?

So we have an angle, a side, and an angle. You want to trace it in one direction. You don't want to jump around.

So therefore, we have the ASA postulate, angle, side, angle. And so we can make the statement that triangle RST is congruent to triangle VXY. So again make sure that the letters match.

So R corresponds to V, S corresponds to X, and T corresponds to Y. So let's look at another example. So let's call this triangle DEF. triangle ABC and let's say that DE is congruent to AB and DF is congruent to AC an angle D is congruent to angle a what postulate can we use to prove that these two triangles are congruent So we have a side, an angle, and a side.

So it's side, angle, side, S-A-S. And so we could therefore say that triangle D-E-F is congruent to triangle A-B-C. And I forgot the triangle symbol. What about this one? Let's call this triangle XYZ and let's say this is ABC.

And let's say that XZ is congruent to AC and angle Z is congruent to angle C. and angle B is congruent to angle Y. What postulate can we use to prove that these two triangles are congruent?

So we have an angle, an angle, and a side. So by the AAS postulate, these two are congruent. So triangle XYZ is congruent to triangle ABC. Now sometimes you may have a composite triangle. That is two triangles within a single triangle.

So let's say that this is A, B, C, and D. And let's say that we're given that AD is congruent to CD, and AB is congruent to BC. What can we prove?

What conclusions can we draw? So let's separate this composite triangle into two triangles. So the first one on the left is triangle ABD.

On the right, we have triangle CBD. So we know that AD and DC are the same, AB and BC are the same, but notice that they share a common side, and that is DB is equal to DB, because that's the same segment. So therefore, we have two triangles that share, that have the same three sides.

So these two triangles are congruent by the SSS postulate. So sometimes if you don't see the two triangles, you may want to split this composite triangle into two smaller triangles, if that helps. So we can make the statement that triangle ABD is congruent to triangle CBD.

Let's try another composite figure. Let's call this A, B, C, D, and E. So let's say that AC is congruent to EC, and also angle A is congruent to angle E.

What can we do to prove that the two triangles are congruent? The triangle on the left, ACB, and the triangle on the right, ECB. Right now we only have two things that are congruent.

We have an angle and a side. Is there anything else that we could see that's congruent? Now whenever you have two lines that intersect each other, or two segments, they will produce something that has vertical angles vertical angles are congruent so in this picture angle one is congruent to angle 2 and 3 is congruent angle 4 so therefore angle a CB is congruent to ECD and we now could prove that the two triangles are congruent by the ASA postulate So we have two angles and the side between the two angles.

So now we can say that triangle ACB is congruent to triangle ECD. Notice that E and A are the same in this particular example. And D and B correspond to each other. So let's say that this is triangle ABC and triangle DEF. Now let's say that angle A is congruent to angle D, and that angle B is congruent to angle E, and angle C is congruent to angle F.

So are these two triangles congruent? so we have an angle and angle and an angle so is there such a possible at the AAA possible no there isn't so therefore these two triangles are not congruent so we can write this statement triangle ABC I put S for some reason. Let's correct that. It is not congruent to triangle D, E, F. So let's say this is X.

YZ, let's call this A. And in this example, we're going to say that angle X is equal to angle Z, and XYA is congruent to ZYA. And with this information, what can we conclude about these two triangles?

Can we say that triangle XYA is congruent to triangle ZYA? Now remember, they share a common side, that is YA. So we could say that YA is congruent to YA for the two small triangles based on a reflexive property.

Thus, we have an angle, an angle, and a side. So by the AAS postulate, which is one of the four that we mentioned in the beginning, these two triangles are congruent. So triangle X, Y, A is congruent to triangle Z, Y, A. So let's say this is Q, R, S, T. Let's say X. And let's say that QS is congruent to ST and RS is congruent to SX. Prove that these two triangles are congruent.

So as you recall, angle RSQ is a vertical angle with XST. Those two are vertical angles, which means that they're congruent. So we have a side, an angle, and a side. So by the SAS postulate... we can say that triangle RSQ is congruent to triangle XST.

So notice that R and X are opposite to the same side. So therefore, R and X are congruent. And T...

is opposite to the same side as Q. So Q and T are congruent. So therefore triangle R S Q is congruent to triangle X S T. Now let's look at one last example before we work on two column proofs.

Let's call this A, B, C, and D. And let's say that AB is congruent to AD, and BC is congruent to DC. What do we need in order to prove that these two triangles are congruent? Now, they share the same side, side AC.

So therefore, we have a side, side, and a side. So by the SSS postulate, we can say that triangle ABC is congruent to triangle ADC. And that's it. Now let's work on some two-column proofs. Let's say this is A, B, C, and D.

And we're given that segment AD is congruent to CD. And also that B is the midpoint of AC. Your task is to prove that the two triangles are congruent.

That triangle ABD is congruent to triangle CBD. So feel free to pause the video if you want to. So let's write our two columns. Statements. and reasons.

So the first thing you should do as always is rewrite what you're given. So we know that AD is congruent to CD, and that's a given. And we know that B is the midpoint Let me just mark that AD is congruent to CD. Now, B is the midpoint of segment AC. So, that's given.

Now, what are some conclusions that we can draw if B is the midpoint? If B is the midpoint, that means that AB is congruent to BC. So, let's write that in step 3. So, AB is congruent to BC. And the reason? Definition of a midpoint.

Now, let's get rid of this stuff. It's just taking up space. Now, what else can we say?

So right now we have two sides. We either need another angle or another side. Now notice that BD is shared between both triangles.

So that's the third side. That's the common side. So we could say that BD is congruent to itself.

Whenever you have a common side, the reason you can always write is the reflexive property. Now what postulate can we use to prove that these two triangles are congruent? So notice that we have a side, a side, and another side.

So we can make the final statement that triangle ABD is congruent to triangle CBD, and the reason for that is the SSS postulate. And... We're using statements 1 that show that these two sides are congruent, and statement 3, which show that AB is congruent to BC, and also statement 4, where BD is congruent to itself. And so we're going to write 1, 3, and 4. And so that's how you can prove that two triangles are congruent. You can use the SSS postulate, SAS, ASA, or AAS.

By the way, I recommend that you pause the video for each of these problems. And it's best if you work on it, because that's the way you're going to learn. Pause the video, try the problem, and then once you get the answer, unpause it to see the solution.

Now, you might have a different way of getting to the answer, so sometimes there's more than one way. So I just want to let you know about that. Now let's go ahead and begin.

So let's call this MOP, and this is going to be R. and we're given that segment RO is perpendicular to segment MP and also segment MO is congruent to segment OP. Your task is to prove that triangle MRO is congruent to triangle PRO. So go ahead and begin. See if you can do this.

So let's write our two-column proof. So as always, what's the first thing that we should do? Remember the first thing you should do is write the given. So first let's start with the fact that segment MO is congruent to segment OP and that's given. And just to keep things in perspective just to make it easier, let's highlight that.

So here's MO and that's congruent to OP. Let's highlight it with tick marks. Now let's move on to statement 2 and that is segment RO is perpendicular to segment MP.

So that's a given. Now this is RO and this is MP. So they're perpendicular.

What conclusions can you draw from perpendicular lines? At what angles do perpendicular lines meet? A perpendicular line will always form a 90 degree angle with the line that intersects with.

So, the angle between RO and MO is 90. The same is true between RO and OP. Perpendicular lines intersect at right angles. So, therefore, we could say that... angle MOR is congruent to angle POR and the reason we can say definition of perpendicular lines or simply that perpendicular lines form right angles Now what else can we say? Now notice that we do have a common side.

So for statement 4 we can say that RO is congruent to RO. And we know the reason for that. That is the reflexive property.

So now, we have a side, we have an angle, and another side. So we can make the statement that triangle MRO is congruent to triangle PRO. And this is based on the SAS side angle side postulate. And so we've used, in this example, statement 1, which showed that MO is congruent to OP. We use number 3. the two angles are congruent, and number 4 for the common sign.

So this is based on statements 1, 3, and 4. And so that's it. Now let's move on to our next example. So let's say this is A, B, C, D, and E.

We're going to call this angle 1, angle 2, 3, and 4. So given that angle 1 is congruent to angle 4, and also that segment AC is congruent to segment EC, prove, go ahead and prove that Triangle ABC is congruent to triangle ECD. So once again, pause the video and go ahead and work on this problem. So hopefully by now, you're getting the hang of how to do these kinds of problems.

So I'm going to give you a lot of practice problems so you can master this topic. So, statement 1, we can say that angle 1 is congruent to angle 4. And that statement was given to us. So now let's mark it on the diagram.

So, angle 1 is congruent to angle 4. now statement to we know that segment AC is congruent to segment EC and that's given to us as well so here is segment music different color this is segment AC and here is easy now what else can we say about the two triangles Now notice that we have vertical angles. Angle 2 is congruent to angle 3. And why can we say that? Why is angle 2 congruent to angle 3? These two are vertical angles. Remember vertical angles are congruent.

So now we have three things. We have an angle, we have a side, I mean a side by the yellow mark there, and we have another angle. So by the ASA postulate, we can say that triangle ABC is congruent.

It's a triangle E, C, D. And the numbers that we use are basically 1, 2, and 3. It's based on statements 1, 2, 3. Alright, let's try another example. So let's say this is A, B, C, D, and E. So we're given that segment AB is congruent to CD. And segment AE is congruent...

to segment DE. And also that angle 1 is congruent to angle 4. And I need to write those angles. This is angle 1, 2, 3, and 4. And 3 is on the inside. Your task is to prove that triangle ABE is congruent to triangle DCE. So go ahead and take a minute and work on that example.

So how should we begin? Well, let's write the two columns, statements and reasons. So the first statement we're going to say is that AB is congruent to CD, which is given.

And as always, we're going to mark this on the graph. So this is AB, and here's CD. Now the second statement is that AE is congruent to DE, which is also given.

And so this is AE, and this is DE. Now we're given that angle 1 is congruent to angle 4. So we're going to write that as well. Angle 1 is congruent to angle 4. So these two are the same.

Now, if those two angles are the same, then the angles that are supplementary to those angles 2 and 3 are the same as well. So in step 4, we can say that angle 2 is congruent to angle 3. And the reason for that is that supplements of congruent angles are congruent. So let's say if angle 1 was 120, angle 4 has to be 120. That means number 2 has to be 60. 60 plus 120 is 180. So 3 is 60, which means 2 equals 3. Now, step 5. We can make the final statement that triangle ABE is congruent to triangle DCE. Now, what postulate can we use to show that those two triangles are congruent?

So, let's see. We have, let's mark that these two are congruent, angles 2 and 3. So we have a side, an angle, and a side. So hopefully you see it.

You can also see on this side too. Side, angle, side. So it's the SAS postulate.

And we've used statements 1, 2, and 4 to accomplish the congruence of these two triangles. So it's 1, 2, and 4. And that's it for this problem. Here's another one that we can work on. Similar to the first two, but with a few differences. So let's call this T, Q, R, and S.

And we're given that angle T, Q, R. is congruent to angle TSR. And we're also given that TR bisects angle QTS. So your task is to prove that triangle QTR is congruent to triangle STR.

So go ahead and work on it. So, number one. The given, TQR is congruent to TSR. So TQR, we could describe it as angle Q. And TSR, we could say is just angle S.

Because angle Q can only correspond to this angle. So I'm going to say angle Q is congruent to angle S. Let's write it.

And the reason is that it's given. now number two we know that ray TR bisects angle QTS so this is QTX and this is ray TR it bisects it or splits it into two congruent angles which means that these two angles are congruent Let's put the reason for statement 2 as given. In statement 3, we can say that angle QTR is congruent to angle STR.

Now, the reason for this that we can write is based on the definition of an angle bisector. This is an angle bisector. Angle bisectors are rays, and they split an angle into two congruent angles.

Now, we do have a common side. So every time you have a common side, you can use the reflexive property. So we're going to state that TR is congruent to itself, TR.

And that's the reflexive property. Now finally, we can make the fifth statement that triangle QTR is congruent to triangle STR. And what postulate can we use to say this? So here we have...

an angle, another angle, and a side. So this is the AAS postulate. And we use number 1 to prove that angle Q is equal to angle S, and number 2, that just helps us to establish number 3 which...

We did use that. That's the second angle. And for the side, that's number 4. So it's based on statements 1, 3, and 4. And that's how you can prove the two triangles, using the AAS postulate, angle-angle-side. Now let's move on to our next problem.

So let's call this A, B, C, D, and E. And you're given that C is the midpoint of two segments. C is the midpoint of A, E.

and it's the midpoint of BD. So with this information, go ahead and prove that triangles ABC is congruent to triangle EDC. So how can we prove that triangle ABC is congruent to triangle EDC? What would you say?

So let's start with the given. So we're given that C... is the midpoint of AE and BD. So that's a given. Now, number two, because C is the midpoint of AE, that means that AE is bisected into two congruent segments.

So we could say that AC is congruent to EC. And the reason is definition of a midpoint. Now, number three, we could say that BC is congruent to CD. Because C is still the midpoint of BD. And the reason for that is the same.

Definition of a midpoint. Now what else can we say? Another statement that we can make... is that angle ACB is congruent to angle ECD.

Now, I want to mention that A corresponds to E, not D in this case. Because look at the angle, or rather, the side that angle A is across. Angle A is across the side with two marks, and angle E is across the side with two marks. So if these two sides are the same, the angles that are opposite to them should be the same.

So that's how we know to match A with E and B with D. So those two angles are congruent because the vertical angles are congruent. So now we have enough information to prove that triangle ABC is congruent to triangle EDC. So we have a side, an angle, and a side. So using the SAS postulate.

So we've used statement 2, 3, and 4 to prove that these two triangles are true. So let's write that here. And that's it for this problem. So let's say that this is ABC and this is going to be DEF and we're given that AD is congruent to CF and also that CB is congruent to DE. and AB is congruent to EF.

Your task is to prove that the two triangles are congruent. So we're going to try to prove that triangle ABC is congruent to triangle FED. So let's start with the fact that AD is congruent to CF and that's given.

And number 2, CB is congruent to DE. So that's given as well. And 3, AB is congruent to EF. Now the reason why I wanted to write all of that at the beginning is to make some more space. So we know that AD is congruent to CF.

Now it might be helpful to draw two separate triangles. So this is going to be triangle ABC and that's triangle FED. Let me use a different color for that. Now, based on statement 2, CB is congruent to DE. So here's CB and here's DE.

Those two sides are congruent. And AB is congruent to EF. So this is AB and this is EF.

So if we can prove that AC is congruent to DF, then we can prove that the two triangles are congruent using the SSS postulate. So how can we prove that these two segments are congruent? What can we do?

So notice that we have an overlapping segment, DC. So that implies we need to use segment addition of some sort. First, we're going to say that AD is equal to CF before we perform some mathematical operation.

We can say this based on definition of congruent segments. To get AC and DF, what I need to do is I need to start with this sentence or this equation and add segment DC to both sides. I'm going to have AD plus DC, and that's equal to CF plus DC.

Now, because I added it to both sides, this is simply the addition property. Now, notice that AD plus DC is AC. So, I can replace the left side with AC.

And DC plus C... is DF so therefore I can replace the right side with DF and whenever you add in two small segments to get a larger segment this is known as segment addition now step 7 I can say that AC is congruent to DF and this is definition of congruent segments so now I'm going to put the mark so this is AC and this is DF so now we can make the final statement that triangle ABC is congruent to triangle F ED And the reason for that is the side, side, side postulate. And so we've used, let's see, number 2, CD is equal to, CB is equal to DE. And we've used number 3, AB is equal to EF.

And also number 7. So it's 2, 3, and 7. And that's it for this problem. Now this is going to be the last problem of the video. So this is going to be A, B, C, D, and E.

And so here's what we're given. We're given that B and C, that is points B and C, trisect segment AD and then we're given that angle A is congruent to angle D and also angle EBC is congruent to angle ECB so go ahead and prove that triangle EAB is congruent to triangle EDC. So take a minute and work on this problem. So let's start with the first one.

Let's say that angle A is congruent. to angle D and that is given. And let's mark it on the graph.

So here's A and here's D. They're congruent to each other. Now number two, we know that angle EBC is congruent to ECB. So this is EBC. So that's congruent to ECB.

Let me just use a different color. Instead of having so many marks. So if those two angles are equivalent, what can we conclude? Remember, our goal is to prove that triangle EAB is congruent to EDC. So we don't need these two angles.

It's outside of the triangles that we want. However, we do want the angles that are supplementary to those angles. So that means that angle EBA is congruent to ECD. because there's supplements of congruent angles.

So let's write that. Statement 3. Angle EBA is congruent to angle ECD. And the reason? Supplements of congruent angles are congruent.

Now, we just need one more piece of information, and it has to do with this. So, let's rewrite that statement. So, number 4, points B and C trisect AD. So, that's given.

And what is the conclusion of that? Well, when something bisects a segment, it splits it into two equal parts. When you have two points that trisect the segment, it splits it into three equivalent parts. So, AB and BC and CD are equal to each other. Bar focus, well we can write all of it.

So AB is congruent to BC, which is congruent to CD. And that's the definition of an angle trisector. Or rather, not an angle trisector, but a segment trisector. Because we're trisecting a segment, not an angle.

So now we have everything we need. We have an angle, a side, and an angle. And the same is true for the other triangle. Angle, side, angle. So now we can make the statement that triangle EAB is congruent to triangle EDC and this is based on the angle side angle postulate and we've used statement one that is a equals D and we also use a statement 3, EBA is equal to ECD and also statement 5 where AB is congruent to CD.

So this is 1, 3, and 5. And so now that is it. So hopefully this video gave you enough practice where you can master the ability of proving if two triangles are congruent. So the next section is C. Cp, Ct, C. Corresponding parts of congruent triangles are congruent. which is based on this video.

So basically, everything you've learned in this video, you're going to need it for the next video, or for the next topic. So once you prove that two triangles are congruent, you can prove that any other part is congruent. For example, in the next section, you just got to take it a step further.

I could say that AE is congruent to ED. And all I gotta say is C-P-C-T-C. If two triangles are congruent, then the corresponding parts are congruent. So that's the next topic though, which you can look for my video on YouTube. Thanks for watching.

Transcript for:Understanding Triangle Congruence Postulates

Transcript for:
Understanding Triangle Congruence Postulates