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 B C a 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 we know that triangle BCD is congruent so BCD is congruent to what we know it's congruent all of these three triangles are congruent to each other so for example BCD is congruent to EC D and so their corresponding sides and corresponding angles will also be congruent so just looking at the order in which they're written be a vertex B corresponds in this triangle and BCD corresponds to vertex B in BC a be CA so this is the B vertex in BC a which corresponds to the eevr techs and EC D so all everything that I've done in magenta all of these angles are congruent and then we also know we also know that the C angle so in BC a is sorry BC D this angle this angle right over here is congruent to the C angle and 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 C angle I guess we could call it in ECE but an EC D 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 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 listed B so in triangle BCD this angle this angle right over here corresponds to the a vertex angle in BCA so BC a 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 congruence YZ and now we can come up with some interesting things about them first of all here angle BCA angle BC D and angle D C II they're all congruent and when you add them up together you get to 180 degrees if you put them all adjacent as they are 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 and 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 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 plus 60 plus something is going to add up to 180 90 plus 60 is 150 so this has to be 30 degrees to end up 280 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 can also think about these outer angles so this or not the outer angles where these combined angles so angle say AC or say angle a b e 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 three smaller triangles are all they all have the exact same angles 30 60 90 and the exact same side lengths width and 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 a B II that's clearly not congruent it's a larger triangle it has a different measures for its length but it has same angles 30 60 and then 90 so it's actually similar to all of the triangles that it's made up of