So we have a parallelogramright end here. And also what I desire to proveis the its diagonals bisect each other. So the an initial thing thatwe can think about-- this aren't just diagonals. These are lines the areintersecting, parallel lines. So girlfriend can also viewthem as transversals. And if we emphasis onDB ideal over here, we check out that itintersects DC and AB. And also since we understand thatthey're parallel-- this is aparallelogram-- we know the alternative interiorangles need to be congruent. So the angle must beequal to that angle there. And also let me do a brand here. Permit me speak to thatmiddle point E. Therefore we understand that angle ABE mustbe congruent to angle CDE by alternating interior anglesof a transversal intersecting parallel lines. Now, if us look atdiagonal AC-- or us should contact it transversal AC--we have the right to make the very same argument. The intersects here and also here. These two lines room parallel. So alternative interiorangles should be congruent. So angle DEC have to be-- so letme write this down-- edge DEC must be congruent come angleBAE, for the precise same reason. Currently we have somethinginteresting, if us look at thistop triangle end here and also this bottom triangle. We have one set of correspondingangles that room congruent. We have actually a next in betweenthat's going to be congruent. Actually, let me writethat down explicitly. Us know-- and also we provedthis to ourselves in the vault video-- thatparallelograms-- not just are opposite political parties parallel,they are also congruent. Therefore we recognize fromthe previous video that the side isequal to that side. Therefore let me go back towhat i was saying. We have two to adjust ofcorresponding angles that room congruent, wehave a side in in between that's congruent, andthen we have another collection of corresponding anglesthat are congruent. Therefore we recognize that this triangleis congruent to that triangle by angle-side-angle. Therefore we recognize thattriangle-- I'm going to go from the blue to theorange come the last one-- triangle ABE is congruent totriangle-- blue, orange, climate the critical one-- CDE, byangle-side-angle congruency. Now, what walk that carry out for us? Well, we understand if twotriangles space congruent, all of theircorresponding features, especially every one of theircorresponding sides, are congruent. For this reason we understand that side ECcorresponds to next EA. Or I might say next AEcorresponds to next CE. They're corresponding sidesof congruent triangles, therefore their steps or theirlengths have to be the same. So AE have to be same to CE. Allow me put two slashessince I already used one slash over here. Now, by the sameexact logic, we know that DE-- let mefocus top top this-- we understand that it is in mustbe same to DE. Once again, they'recorresponding sides of 2 congruent triangles, sothey must have actually the very same length. So this is correspondingsides of congruent triangles. So be is same to DE. And also we've excellent our proof. We've displayed that, look,diagonal DB is dividing AC right into two segments of equallength and also vice versa. AC is dividing DB into twosegments of same length. Therefore they arebisecting every other. Now let's go theother method around. Let's prove toourselves the if we have actually two diagonals ofa quadrilateral that are bisecting eachother, the we are dealing witha parallelogram. So let me see. So we're going to i think thatthe 2 diagonals are bisecting each other. So we're suspect thatthat is same to that and that that right overthere is equal to that. Provided that, we want to provethat this is a parallelogram. And to perform that, we justhave to remind ourselves the this angle is walk tobe same to the angle-- it's among the an initial things welearned-- since they space vertical angles. So let me write this down. Allow me label this point. Angle CED is goingto be equal to-- or is congruent to-- angle BEA. Well, that reflects usthat these 2 triangles are congruent because we haveour matching sides that room congruent, an angle inbetween, and then another side. So we now recognize thattriangle-- I'll store this inyellow-- triangle AEB is congruent come triangle DECby side-angle-side congruency, by SAS congruent triangles. Same enough. Now, if we understand that twotriangles are congruent, we know that all of thecorresponding sides and angles are congruent. So because that example, weknow that angle CDE is going come becongruent to edge BAE. And this is simply correspondingangles that congruent triangles. And now we have thistransversal of these two lines that can be parallel, if thealternate interior angles space congruent. And we check out that castle are. These 2 are sort of candidatealternate internal angles, and they are congruent. So abdominal muscle must it is in parallel come CD. Actually, I'll justdraw one arrow. Abdominal muscle is parallel come CD byalternate interior angles congruent of parallel lines. I'm just writingin part shorthand. Forgive the crypticnature of it. I'm speak it out. And also so we have the right to thendo the exact same-- we've just presented that thesetwo sides space parallel. We might then dothe specific same reasonable to show that this twosides space parallel. And also I won't necessarilywrite it every out, but it's the precise sameproof to show that this two. So first of all, weknow that this edge is congruent come thatangle best over there. Actually, allow me create it out. For this reason we know that angle AECis congruent to angle DEB. They are vertical angles. And also that to be our reasonup here, together well. And also then we watch thetriangle AEC must be congruent come triangleDEB by side-angle-side. For this reason then we havetriangle AEC must be congruent to triangleDEB through SAS congruency. Climate we know that correspondingangles must be congruent. So for example, angle CAE mustbe congruent to angle BDE. And also this is they'recorresponding angles of congruent triangles. For this reason CAE-- let me dothis in a brand-new color-- should be congruent to BDE. And also now we have a transversal. The alternative interiorangles are congruent. For this reason the 2 lines that thetransversal is intersecting need to be parallel. Therefore this have to beparallel come that. So climate we have actually ACmust be parallel to it is in BD by alternative interior angles.


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And also we're done. We've just proven thatif the diagonals bisect each other, if we start that asa given, climate we end at a allude where we say, hey, the oppositesides the this quadrilateral need to be parallel, or thatABCD is a parallelogram.