Determining congruent triangles example

We have four triangles depicted here, and they've told us that the triangles are not drawn to scale. We are asked which two triangles must be congruent. So, pause this video and see if you can work this out on your own before we work through this together.

All right, now let's work through this together. It looks like for every one of these, or actually almost every one of these, they've given us two angles, and they've given us a side. This triangle, I J H, they've only given us two angles. So what I'd like to do is, if I know two angles of a triangle, I can figure out the third angle because the sum of the angles of a triangle have to add up to 180 degrees. Then I can use that information, maybe with the sizes that they give us, in order to judge which of these triangles are congruent.

So first of all, what is going to be the measure of this angle right over here, the measure of angle A C B? Pause the video and try to think about that.

Well, one way to think about it is: if we call the measure of that angle x, we know that x plus 36 plus 82 needs to be equal to 180. I'm just giving their measures in degrees here. So, you could say x plus... let's see, 36 plus 82 is 118. Did I do that right? Six plus two is eight, and then three plus eight is eleven. Yep, that's right. So, that's going to be equal to 180. Then, if I subtract 118 from both sides, I am going to get x is equal to 180 minus 118, which is 62. So this is x is equal to 62, or this is a 62 degree angle—I guess is another way of thinking about it. I could put everything in terms of degrees, if you like.

All right, now let's do the same thing with this one right over here. Well, this one has an 82 degree angle and a 62 degree angle, just like this triangle over here. So we know that the third angle needs to be 36 degrees, because we know 82 and 62. If you need to get to 180, it has to be 36. We just figured that out from this first triangle over here.

Now, if we look over here, 36 degrees and 59, this definitely looks like it has different angles, but let's figure out what this angle would have to be. So, if we call that y degrees, we know... I'll do it over here: y plus 36 plus 59 is equal to 180. I'm just thinking in terms of degrees here. So y plus... this is going to be equal to what is this? This is going to be equal to 95, is equal to 180. Did I do that right? Yep, that's 80 plus 15, you have 95. And then, if I subtract 95 from both sides, what am I left with? I'm left with y is equal to 85 degrees. And so this is going to be equal to 85 degrees.

And then this last triangle right over here, I have an angle that has a measure of 36, another one that's 59. So by the same logic, this one over here has to be 85 degrees.

So, let's ask ourselves now that we've figured out a little bit more about these triangles, which of these two must be congruent? You might be tempted to look at these bottom two triangles and say, "Hey, look, all of their angles are the same." You have angle-angle-angle and angle-angle-angle. Well, they would be similar if you have three angles that are the same; you definitely have similar triangles.

But we don't have any length information for triangle I J H. You need to know at least one of the lengths of one of the sides in order to even think—start to think about congruence. So we can't make any conclusion that triangle I J H and triangle L M K are congruent to each other.

Now, let's look at these candidates up here. We know that their angles are all the same, and so we could apply... we could apply angle-side-angle: 36 degrees, length 6; 82 degrees; 36 degrees, length 6; 82 degrees. So by angle-side-angle, we know that triangle A B C is indeed congruent to triangle D E F, and we're done.