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# Proof: Diagonals of a parallelogram

CCSS.Math:

## Video transcript

so we have a parallelogram right over here what I want to prove is that its diagonals bisect each other so the first thing that we can think about these aren't just diagonals these are lines that are intersecting parallel lines so you can also view the Miss transversals and if we focus on D B right over here we see that it intersects DC and a B and since they're those we know our parallelogram we know that they're parallel this is a parallelogram we know that alternate interior angles must be congruent so that angle must be equal to that angle there and let me make a label here let me call that middle Point E so we know that angle a B E we know that angle a B e must be congruent to angle CD to angle C de by alternate interior angles of a transversal intersecting parallel lines and alternate interior alternate interior angles now if we look at if we look at diagonal AC or we should call it transversal a so you can make the same argument it intersects here and here these two lines are parallel so alternate interior angles must be congruent so angle Dec must be so let me write this down angle Dec must be congruent to angle BAE BAE to angle B angle BAE by for the exact same reason now we have something interesting if we look at this top triangle over here in this bottom triangle we have one set of corresponding angles that are congruent we have we have a side in-between that's going to be congruent actually let me write that down explicitly we know we know then we prove this to ourselves in the previous video that parallelograms not only are opposite sides parallel they are also congruent so we know from the previous video that that side is equal to that side so let me go back to what I was saying we have two sets of corresponding angles that are congruent we have a side in between that's congruent and then we have another set of corresponding angles that are congruent so we know that this triangle is congruent to that triangle by angle side angle angle side angle so we know that triangle I'm going to go from the blue to the orange to the last one triangle a B E is congruent to triangle blue orange in the last one CDE CDE by angle-side-angle congruency angle-side-angle congruence now what does that do for us well we know if two triangles are congruent all of their corresponding features especially all the corresponding sides are congruent so we know that side we know that side EC side EC corresponds to side E a or I could say side AE we could say side a correspond a corresponds to side C e to see e their corresponding sides of congruent triangles so they're measures or their lengths must be the same so AE must be equal to C let me put two slashes it's already used one slash over here now by the same exact logic we know that de we know that let me to focus on this we know that be e we know that be e must be equal to de be e must be equal to de once again so their corresponding sides of two congruent triangles so they must have the same length so this is corresponding sides of congruent congruent triangles so be e b e is equal to de and we've done our proof we've shown that look diagonal DB is splitting AC into two segments of equal length and vice versa ac is splitting DB into two segments of equal length so they are bisecting each other now let's go the other way around let's prove to ourselves that if the if we have two diagonals of a quadrilateral that are bisecting each other that we are dealing with a parallelogram so let me see so we're going to assume that the two diagonals are bisecting each other so we were assuming that that is equal to that and that that right over there is equal to that given that we want to prove that this is a parallelogram and to do that we just have to remind ourselves we just have to remind ourselves that this angle is going to be equal to that angle it's one of the first things we learned because they are vertical angles so let me write this down see label this point CED angle C II D is going to be equal to or is congruent to angle so I started C is B EA angle B e a and that what is that well that shows us that these two triangles are congruent because we have a corresponding size if it congruent an angle in between and then another side so we now know that the triangle keep this in yellow triangle a a EB is congruent to triangle a B is congruent to triangle d ee c d e c bi side-angle-side congruence side-angle-side congruence by SAS congruent congruent triangles fair enough now if we know that two triangles are congruent we know that all of the corresponding sides and angles are congruent so for example we know that angle CDE angle CDE angle C D E is going to be congruent to angle you can see CDE is going to be congruent to BAE BAE is going to be congruent to angle B to angle BAE and this is just corresponding angles corresponding angles angles of congruent of congruent triangles and now we have we have this transversal or this can't this kind of transversal of these two lines that could be parallel if the alternate interior angles are congruent and we see that they are these two are kind of candidate alternate alternate interior angles and they are congruent so a B must be parallel to CD so a bee she'll just draw one arrow a bee must be parallel to CD a bee is parallel to CD by alternate interior angles congruent of parallel lines I'm just writing in some shorthand forgive the cryptic nature of it LM saying it out and so we can then do the exact same log we've just shown that these two sides are parallel we can then do the exact same logic to show that these two sides are parallel and I won't necessarily write it all out but it's the exact same proof to show that these two so first of all we know that this angle is congruent to that angle right over there and then we know actually let me write it out so we know that angle AEC angle a EC is congruent to angle D let me EB I should say de be angle D e be they are vertical angles vertical angles and that was our reason up here as well vertical vertical angles and then we see that triangle AEC must be congruent to triangle de B by side angle side so then we have triangle AEC must be congruent to triangle D EB by s.a.s congruence then we know that corresponding angles must be congruent so that we know that angle so for example angle c AE angle c AE must be congruent to angle BD e BD e angle b d e and this is their corresponding angles of congruent triangles so c AE c AE let me do this in a new color c AE c AE must be congruent to be to be de b d e and now we have a transversal the alternate interior angles are congruent so the two lines that the transversal is intersecting must be parallel so this must be parallel to that so that we have AC AC must be parallel to BD BD by alternate interior angles alternate interior angles and we're done we've just proven that if the diagonals bisect each other if we start that as a given then we end at a point where we say hey the opposite sides of this quadrilateral L or that ABCD is a parallelogram