Working with triangles
So we have this larger triangle here and inside of that, we have these other triangles, and we're given this information right over here. That triangle BCD is congruent to triangle BCA, which is congruent to triangle ECD. And given just this information, what I want to do in this drawing, I want to figure out what every angle on this drawing is. What's the measure of every angle? So let's see what we can do here. So let's just start with the information that they've actually given us. So we know that triangle BCD is congruent to-- well, we know all of these three triangles are congruent to each other. So, for example, BCD is congruent to ECD, and so their corresponding sides and corresponding angles will also be congruent. So just looking at the order in which they're written B, vertex B corresponds, in this triangle, BCD, corresponds to vertex B in BCA, so this is the B vertex in BCA, which corresponds to the E vertex in ECD. So all-- everything that I've done in magenta, all of these angles are congruent, and then we also know that the C angle. So in BCA-- sorry, BCD, this angle right over here, is congruent to the C angle in BCA. BCA, the C angle is right over here, or C is the vertex for that angle in BCA. And that is also the C angle, I guess we could call it, in ECD. But in ECD, we're talking about this angle right over here. So these three angles are going to be congruent. And I think you could already guess a way to come up with the values of those three angles. But let's keep looking at everything else that they're telling us. Finally, we have vertex D over here. So angle-- so this is the last one in where we listed-- so in triangle BCD, this angle right over here corresponds to the A vertex angle in BCA. So BCA, that's going to correspond to this angle right over here. It's really the only one that we haven't labeled yet. And that corresponds to this angle, this vertex right over here, that angle right over there. And just to make it consistent, this C should also be circled in yellow. And so we have all these congruences, and now we can come up with some interesting things about them. First of all, here, angle BCA, angle BCD, and angle DCE, they're all congruent, and when you add them up together, you get to 180 degrees. If you put them all adjacent, as they all are right here, they end up with a straight angle, if you look at their outer sides. So you have, if these are each x, you have three of them added together have to be 180 degrees, which tells us that each of these have to be 60 degrees. That's the only way you have three of the same thing adding up to 180 degrees. Fair enough. What else can we do? Well, we have these two characters up here. They are both equal and they add up to 180 degrees. They are supplementary, the only way you can have two equal things that add up to 180 is if they're both 90 degrees. So these two characters are both 90 degrees. Or we could say this is a right angle, that's a right angle. And this is congruent to both of those, so that is also 90 degrees, and then we're left with these magenta parts of the angle. And here, we could just say, well, 90 plus 60 plus something is going to add up to 180. 90 plus 60 is 150. So this has to be 30 degrees to add up to 180. And if that's 30 degrees, then this is 30 degrees. And then this thing right over here is 30 degrees. And then the last thing-- we've actually done what we said we would do, we found out all of the angles. We could also think about these outer angles. So this-- or not the outer angles, or these combined angles. So angle say AC-- or say, angle ABE, so this whole angle we see is 60 degrees. This angle is 90 degrees, and this angle here is 30. So what's interesting is these three smaller triangles, they all have the exact same angles, 30, 60, 90, and the exact same side lengths. We know that because they're congruent. And what's interesting is when you put them together this way, they construct this larger triangle, triangle ABE, that's clearly not congruent. It's a larger triangle. It has different measures for its lengths, but it has the same angles 30, 60, and then 90. So it's actually similar to all of the triangles that it's made up of.