Corresponding angles in congruent 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 wanna do in this drawing, I wanna 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, we know that triangle BCD is congruent so BCD is congruent to, well we know it's congruent all of these 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 word written, B, vertex B corresponds, in this triangle, in BCD , corresponds to vertex B in BCA, BCA So this is the B vertex in BCA which corresponds to the E vertex in ECD So all everything that I've done magenta all of these angles are congruent And then we also know, we also know that the C angle, so in BCA sorry, BCD This angle, 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, the, the C angle I guess you 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 I think you could already guess a way to come up with the values of those three angles But let's just let's keep looking at everything else that they're telling us Finally, we have vertex D over here So angle so, angle so this is the last one in, where we lifted, B so in triangle BCD, this angle, this angle right over here corresponds to the A vertex angle in BCA So BCA, that corresponds to this angle right over here It's really the only one that we haven't labelled 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 congruencies 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, cause they all are right here, they'd 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 And the only way we can have two equal things that add up to 180 is that 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, that is also 90 degrees And then we're left with these magenta parts of the angle And here we can just say, well 90 + 60 + something is going to add up to 180 90 + 60 is 150, so this has to be 30 degrees to add up to 180 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've found out all of the angles We can also think about these outer angles, so this or not outer angles, or just combined angles So angles, say AC, oh sorry angle ABE So this whole angle we see is that whole angle is 60 degrees This angle is 90 degrees And this angle, here, is 30 So what's interesting is, these smaller triangles are all they all have the exact same angles: 30, 60, 90, and the exact same side lengths We know that because they're congruent But 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 that has different measures for its lengths, but 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