 Welcome friends to this session on properties of a parallelogram and we will be discussing in subsequent few sessions a lot of theorems related to parallelograms and quadrilateral in general. So it's always advisable to make a list of all these theorems and keep it handy whenever you are solving problems on quadrilaterals. This way you don't need to mug up any particular theorem and by repetition of the uses of all these theorems it will be by hearted automatically. So let's begin. So in this session we are going to discuss theorem one, the first theorem which is it says a diagonal of a parallelogram divides it into two congruent triangles. So the strategy which we will be adopting here is we will be also revising the properties of parallelograms that is we will be restating all those properties and then see how best we can use those properties. So a diagonal of a parallelogram so you know what a diagonal is so a line segment joining the opposite vertices of a quadrilateral is called a diagonal so in this case you can see clearly BD is a diagonal. Now ABCD is a parallelogram so let's start our proof. So we are saying ABCD is a parallelogram is a parallelogram. So what does a parallelogram mean what are the properties? So that means and that means AB is equal to DC as well as parallel to DC. Similarly AD is parallel to BC so this is AB parallel to BC and AB is equal to or sorry AD is equal to BC. So we have to prove that triangle ADB is congruent to triangle CBD. Let's do it methodically. So given is ABCD is a parallelogram and we have to prove to prove diagonal, diagonal BD divides the two sorry divides the parallelogram divides the parallelogram parallelogram parallelogram into two congruent triangles, congruent triangles. This is what is to be proved okay so let's start the proof let's begin. So it's given that ABCD is a parallelogram therefore we can write AB is parallel to CD or DC whichever AB is parallel to CD because ABCD is a is a parallelogram correct. So this is the first step we know. So AB is parallel to CD hence hence and also you can write also AD is parallel to BC same reason therefore we can write angle ABD is equal to angle ABD that means this angle is equal to this angle angle BDC by alternate interior angles. And why do we require all this sooner C so we have to prove that ADB is congruent to CBD. So before that let's first meet all the requirements so hence ADC is also if you see ADC sorry ADB not C ADB is equal to angle CBD isn't it let me mark it as well so this is ADB and this is equal to CD CBD okay again same reason what alternate interior angles I am writing in shorthand. Now let we have to prove that the diagonal is dividing the quadrilateral into two congruent triangles. So we will say in triangle ABD and triangle CDB please be very very mindful of the congruence part what do I mean so if you see if you have to prove that they are equal you have to have corresponding sides equal as well so if you see here I am writing A corresponding to C that means A must be equal to C angle A must be equal to C similarly angle B must be equal to D which is likely to be proved later and D should be equal to B right angle that means the vertices and the angle correspondence must be there so let's see now in triangle ABD and CDB if you see angle ABD is equal to angle BDC proved above proved above here in say one let's say this is two ABD is equal to DDC also DB is equal to DB common side common side and angle ADB will be equal to angle CBD again proved above in two if you look at two equation number two we just proved it okay so now I am writing it here so what is it that means triangle ABD is congruent to triangle CDB okay there is a chance that you might make an error here so please be very very careful in terms of what all correspondence you are making so for example if you see what do I mean by this I mean that if this is angle ABD or here look at this point AB right angle ABD must be equal to angle CDB so is is that so ABD is equal to ABD is equal to CDB yes we just proved this so that is correspondence now in the triangle ABD the vertex or at the angle at D so if you see in triangle ABD angle at D let me use this highlighter so this is an angle I am talking about now this angle must be equal to angle B in CDB so this angle yep so correspondence correspondence is achieved similar and then angle A must be equal to angle C which is true and if you see side DB where is side DB if you side see this is BD here and DB here so both are equal actually so if you see BD side is equal to DB side so hence all correspondence of parts are there so hence you have to have very very clear understanding about it so let's say you what would be the mistake like mistakes would be like triangle ABD many times people write triangle CDB now this is a wrong statement to make wrong wrong why because angle A is angle equal to C but angle B is not equal to angle B in the corresponding triangles for example we are talking about this triangle so this triangle angle B is something like this and and in this triangle angle B is something like that now this angle here is not equal to this angle here correct so hence correspondence is lost actually this angle here is equal to this angle here so hence B in whatever position you are writing B in the second triangle D should be in the same position so for example here in the first triangle B is in the second position and so hence D should also be in the second position here so that's what I mean when I say correspondence right so this is how you have to solve it so hence this was wrong so you just eliminate it is wrong so hence the true result is this and hence proved hence proved so you have to have very clear understanding of what is correspondence and how to use the properties of parallelogram to prove this first theorem and as I suggested please mention all or please maintain a list of all theorems at one place and use it to your advantage