 This last proof is dealing with something similar to what we've seen before but in a slightly different way here We've got a bisector. However. It's not a segment bisector. It's an angle bisector. I see an bisects angle FAC. Well, here's angle FAC and an cuts it in half and so that means that these angles will be congruent Furthermore, we've got these congruence marks and so that means we've got segment FA and segment CA are congruent and lastly, there's that shared segment AM and So it's getting kind of busy in there, but we've got a pair of sides the blue shared side We've got a pair of angles that come from the angle bisector and Then the green sides FA and CA So we can prove that these two triangles are congruent using side angle side. So let's get to it So we're given that AM bisects FAC and that gave us that these two these two angles were congruent Screen we also had the shared segment and then also these congruence marks Tell me that FA and CA are congruent So in the previous slide, we agreed we were going to use side angle side to prove that the triangles were congruent AS and We have let's see. We've got the red share the red sides that we just mentioned There's the green angles that come from the angle bisector and lastly the shared side Hopefully this stuff is starting to look familiar to you now. So the red sides that was FA and CA and from those congruence marks on the drawing, that's just given information for us The angles the green angles let's call them angle FAN and CAN The fact that those two angles are congruent was established from this given piece of information AN bisects angle FAC and what does bisect mean? It means cut in half So we need that as a statement and then just like in a previous proof the reason that we're going to use To make that leap is essentially the definition on a bisector However, this one is slightly different in previous proofs. We had the segment bisector definition While we're bisecting an angle this time. So what does it mean to bisect an angle is according to the angle bisector definition? And then that shared side is starting to look familiar to us hopefully And then lastly, let's make sure we proved it what it is. We're trying to prove we wanted to prove that segments FN and NC In other words, these little segments are congruent which we can prove using the CPCTC So here we have established that the two triangles themselves are congruent And so therefore all of their parts will be congruent as well And we mentioned that before that's the CPCTC Or if you'd rather the CPCTC is a conditional statement If you'd rather say if the triangles are congruent Then all the parts are congruent. So if the triangles are congruent then all the corresponding parts are congruent So the blue statement is a little bit longer. That's why we can use the CPCTC is an abbreviation of that statement