Information for congruency

So, I have two triangles depicted here and we have some information about each of those triangles. We know that this side of this left triangle has length eight. We know that this side has length seven, and then we know that this angle is 50 degrees.

On this triangle, we see some things that look a bit a little bit familiar. This triangle, this side has length eight. This side has length seven, and this angle right over here has a measure of 50 degrees.

So, my question to you is: Can you definitively say, not assuming that these are drawn to scale because they actually aren't, can you definitively say that these triangles are congruent? Or could you definitively say that they aren't congruent? Or can you not say either? Would you have to say that there's not enough information?

Pause this video and think about that.

So essentially, what we have here are two pairs of sides that have the same length and an angle, but that angle is not between those two sides. If the angle were here and here, then we could use side angle side or side angle side to deduce that, hey, these are congruent. But that's not what we're dealing with; we are dealing with side side angle versus side side angle.

I'm saying the side and the side before the angle because otherwise, if I don't do that, it becomes a little bit crass. So, we're really saying a side side angle is not sufficient to prove congruency.

The reason why it's not is that you can actually construct different triangles with the same constraints. For example, on this rightmost triangle, it could look like this, or it could look like this. The seven side could go down like this and intersect just like that.

Now, you might be saying, "Hey, that's not what it looks like." It looks very similar, but remember we're not going on looks; we have to go based on the information they've given us.

So, you could just as easily, based on the information and the constraints they've given us, have a triangle like this. The very fact that you can create two different triangles that are clearly not congruent based on the exact same information and the exact same constraints tells you that that information, those constraints, are not enough to tell you that these are congruent triangles.