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# Geometry proof problem: midpoint

CCSS.Math:

## Video transcript

so we have these two parallel lines line segment a B and line segment CD they are parallel I should say they're parallel line segments and then we have these transversals that go across them so you have this transversal BC right over here and you know this transversal ad and what this diagram tells us is that the distance between a and e this little hash mark says that this line segment is the same distance as the distance between e and D or another way to think about it is that point E Point E is at the midpoint or is the midpoint of line segment ad and what I want to think about into this video is is point E also the midpoint of line segment BC so we are this is the question right over here so is is e the midpoint the midpoint of of line segment BC and you can imagine based on a lot of the videos we've been seeing lately maybe it has something to do with congruent triangles so let's see if we can set up some congruence relationship between the two obvious triangles in this diagram we have this triangle up here on the left and we have this diagram down here this one kind of looks like it's pointing up this one looks like it's pointing down so there's a bunch of things we know about vertical angles and and angles of transversals the most obvious one is that we have this vertical we have we know that angle a B is going to be congruent or it's measure is going to be equal to the measure of angle CED so we know we know that angle AE be angle a/e the is going to be congruent a eb is going to be congruent to angle Dec to congruent to angle Dec which really just means they have the exact same measure and we know that we know that because they are vertical angles vertical they are vertical they are vertical angles now we also know we know that a B and C D are parallel so this line right over here this is transversal so we know for example and there's actually several ways that we can do this problem but we know that this is a transversal and there's a couple of ways to think about it right over here so let me just continue the transversal so we get to see all of the different angles you could say you could say that this angle this angle right here angle a B angle a B II so this is it's measure right over here you could say that it is the alternate interior angle to angle ECD to this angle right over there and if you didn't just if that didn't jump out of view you would say that the corresponding angle to this one right over here is this angle right up here if you were to continue this line off a little bit these are the corresponding angles and then this one is vertical but either way angle AE B let me write this down angle sorry angle a B let me be clear careful angle a the e is going to be congruent to angle so that's a B e is congruent to angle D see E it's congruent to angle D see E and we could say because it's alternate interior angles alternate I'll just write a little code here so alt alt interior alt interior angles and then we have an interesting relationship we have an angle congroo it to an angle another angle congruent angle and then the next side is congruent to the next side over here so pink green side pink green side so we can employ AAS angle angle side and it's in the right order so now we know that triangle we have to make sure that we get the letters right here that we have the right corresponding vertices we can say that triangle triangle a EB triangle a II actually let me start with the angle just to make it interesting angle B ei so we're starting with the magenta angle going to the green angle and then going to the one that we haven't labeled so angle B ei we can say is congruent to angle we start with the magenta and vertices C go to the center II and then go to the unlabeled one D and we know this because of angle angle side and all of and they correspond to each other magenta green side magenta green side they're all congruent so this is from a a s and then if we know that they are congruent that that means corresponding sides are congruent so then so then we know that this side so we know these two triangles can congruent so that means that their corresponding sides are congruent so then we know that length of B E that we know that B e the length of that segment B E is going to be equal and that's the segment that's between the magenta and the green angles the corresponding side is side C e between the magenta and the green angles is equal to C E and this just comes out of the previous statement if we number them that's one that's two and that's three and so that is comes out of statement three and so we have proven us E is the midpoint of BC it comes straight out of the fact that B E is equal to C E so I can mark this off with hash this line segment right over here is congruent to this line segment right over here because we know that those two triangles are congruent and I've inadvertently right here done a little two column proof this over here on the left hand side is my statement statement statement and then on the right hand side I gave I gave my I gave my reason and and we're done