[0:01]In this video, we're going to focus on proven 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.
[1:02]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 could 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 it has to 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.
[2:10]All right, so now let's look at another example. So consider these two triangles, and let's call it triangle RST, and VXY.
[2:27]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.
[3:30]So let's look at another example.
[3:36]So let's call this triangle DEF, and triangle ABC.
[3:50]And let's say that DE is congruent to AB, and DF is congruent to AC, and 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, SAS.
[4:15]And we could therefore say that triangle DEF is congruent to triangle ABC.
[5:26]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 can we, 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.
[7:03]Let's try another composite figure.
[7:08]So let's say this is A, B, C, D, and E. So let's say that AC is congruent to EC. And also that 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.
[7:36]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 known as vertical angles. Vertical angles are congruent. So, in this picture, angle one is congruent to angle two, and three is congruent to angle four.
[8:10]So, therefore, angle ACB 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 a 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 corresponds to each other.
[8:53]So let's say this is A, B, C, and D.
[28:31]Now, if those two angles are the same, then the angles that are supplementary to those angles, two and three, are the same as well. So, in step four, we could say that angle two is congruent to angle three. And the reason for that is that supplements of congruent angles are congruent. So, now we have enough information to prove that triangle ABC is congruent to triangle EDC.
