 Hello and welcome to another session on parallelograms and quadrangles. So we are going to take up another theorem in this session And you can see on the screen the theorem says in a parallelogram the bisectors of any two consecutive angles intersect at right angles, right? So this is a theorem related to parallelogram And they're saying the bisectors of any two consecutive angles, you know what a consecutive angle is even if you don't know Let's try to construct this parallelogram Then take two consecutive angles and try to bisect them And then let us see whether they actually intersect at right angles. So what I'm going to do is I'm going to first draw a Quadrilateral, okay, so let me draw a polygon And that and that polygon is here. So let me start with this point And this point and here c and d. So I think I have drawn a parallelogram Looks like a parallelogram is it and you can see that The the lines are parallel So you don't believe me. Otherwise, you can just draw it In the geojibra software and try for yourself This is a parallelogram. Okay. Now what all are consecutive angles in this so what are they saying they have they are saying that We have to take two consecutive angles. So here there are four pairs of consecutive angles, but all So angle a angle b is one so angle a and angle b So this is one on you know pair of consecutive angles similarly angle b and angle c Similarly angle c and angle d and then angle d and angle a So these are four pairs of consecutive Angles these are Isn't it now they're saying you have to take any two consecutive angle any pair Then try to bisect both the angles in that pair. So let's do that So let me delete this and now I'm going to bisect any two. So let us say I'm taking angle b and angle c Okay, so here is angle b Okay, here is angle b for that matter. So let's say this is angle b This is angle c and I'm going to bisect both of them. So how to do that. There is a tool available in geojibra So I'll draw it and first we'll try to validate it for you And then we'll prove the theorem So here is angle a b c. So let me just bisect it this one and Yeah, so you can see there are two lines which are created now both of them Are the angle bisectors one is internal angle bisector another is external angle bisector So I'm not interested in this. There are lots of interesting theorems around an exterior angle bisectors as well But I am not interested in the exterior angle right now I am just uh interested in bisecting the interior angle. So this is one more. This is one more Yeah, okay. So again, this one is not needed This one is not needed and Here is the point of intersection, right? So e is the point of intersection of angle bisectors of angle b and angle c okay Now you'll be uh thinking that what all these long lines are. So let me do one thing I will You know make them disappear and I will simply Sorry, I disappeared. You know, I did something wrong, huh? So hence I will disappear make this disappear, right this line this line is not needed And I will simply join this with the segment what all so here b and e and e and c Right. So this is what they're saying. So b e is the angle bisector of angle b here E c is the angle bisector of angle c here We have to prove that angle b e c. So let us try to write what we need to prove So we are going to prove Before that we will we will write what is given. So given is What is given guys? So a b c d is a parallelogram, right? a b c d is the parallelogram so in short I'm writing like this and b e and c e are bisectors Bisectors of Angle b and angle c Respectively. Okay. This is what it is. So b e and c e are bisectors of angle b and c respectively therefore Then what we have to prove that to prove To prove angle b e c is equal to 90 degrees That is what they're saying. Why because they are intersecting at right angles meaning what they have to make 90 degrees and point of intersection So what is 90 degrees this angle here? This is 90 degrees This is what they're asking us to prove. So before we prove. Let us see whether it is actually true so let me just take away this part and I'm going to measure this angle. So how do I measure angle b e c so let's say b and e and c Yeah, indeed the software is saying at least that it is 90 degrees. See you can see that But how do we prove that for any? any such Parallelogram, it will be true. Will that be true? So let's try to Shift the parallelogram a bit change the parallelogram. Yes. Now again, I have made another parallelogram. You can see The point of intersection of the angle bisectors are meeting at right angles, isn't it? So let me change it once again. So This is another parallelogram. This is another parallelogram. See again 90 degrees Correct. So whichever parallelogram in fact, there will be special cases. Let's say if it is a rectangle Then also it is 90 degrees. So rectangle, you know, or this in this case, it's a square So even if for rectangle, it will be true. So whichever Quadrangle you take a parallelogram you take and you take any two adjacent angles and then you bisect them And then let the bisectors meet at point e Then you will see e is at right angle If a b c b is a parallelogram. So that is what we learned now let us go back to the original shape or Original case. So this was the case. I think oh no, it was somewhere here. Okay. No problem. So let me take this case a b c b As a parallelogram. Now, let us prove it. So how to prove Let us go to proving part of it. So let's prove So what's the proof? So you know in a parallelogram. We know what do we know? We know that Uh in a parallelogram in a parallelogram adjacent angles are supplementary, isn't it? adjacent angles are supplementary Supplementary Right, what is supplementary angles guys supplementary meaning the sum of two angle has to be 180 degrees Then we say they are supplementary. Now, let's do let's do a sometime some bit of writing work here So let me say this angle is x In value. So this angle is also going to be x y because b e is up Is a bisector of angle b. Now, let us say this angle is y And this angle is y is where right and let us write it also. So we know that since since b e bisects So we'll write like this b e bisects angle b Therefore and let me start right here to save some space. So therefore we can say so guys you have to come from here to this part Okay, so therefore I say angle a b e Is equal to angle c b e Is equal to x. Okay, similarly you can write similarly. What is that? uh angle d c e so angle d c e will be angle b c e And that is equal to y same reason because ec bisects angle c, isn't it? So now what can I say now in triangle? Let us say in triangle e b c Can't I say x plus y plus angle e Is equal to 180 degrees and why is that? This is angle sum property angle sum property of triangle Isn't it guys? We know that the sum of these three angles of a Uh Triangle will be 180 degrees. So hence can I not write x? So this implies x if you see here What is x if I have to write x is nothing but angle b by 2 because 2 times x will be angle b and similarly y will be angle c by 2 Is it so I can write here angle b uh By 2 plus angle c by 2 plus angle e is equal to 180 degrees Now uh So From this particular statement, we can write angle b plus angle c is 180 degrees Isn't it? So angle b plus angle c by 2 will be 90 degrees. So divide this This particular equation by two both sides. So you will get this yes or no. Yeah So let me now write On this side. Okay. So b plus c is 90 degrees. So hence You can say angle e Is equal to 180 degrees minus angle b Plus angle c upon 2 So I have just taken this part To the side right and hence it becomes minus. So hence I can write 180 degrees minus 180 degrees by 2 or which is 90 degrees. Anyways, we had calculated So this is 180 degrees minus 90 degrees. So hence it is 90 Degrees, isn't it? So hence, what do we get guys angle e comes out to be 90 degrees And this is what we wanted to prove and hence we have proved it, right? So once again repeat the theorem the theorem says in a parallelogram the bisectors of any two consecutive angles intersect at right Angles, isn't it? I hope you understood the theorem Now the good The question would be is the converse true Is the converse true that is if let's say, um, you know There is a parallelogram and the two bisectors of consecutive angles are meeting at right angles Would that mean that? the The triangle or the pal What do you say that? Quarter angle is a parallelogram. Is that always true need not be need not be why I will just tell you So in this case the converse is not true. Why because let us say this is another point Let's say d dash And you simply join a and d dash Okay, without doing anything you just simply join a and d dash now consider the quadrangle a b c d dash Now clearly a b c d dash is not a quadrangle, correct? You can see from the figure anyways, it is not a quadrangle, but We have not touched the property that b bisectors of these two consecutive angles Of the quadrangle and meeting at 90 degrees so converse is What would what would be the con converse statement of this theorem the converse statement would have said that even if in a quadrangle The bisectors of two consecutive angles bisect or meet At 90 degrees then the quadrangle or the quadrilateral Is a parallelogram which we prove that no it's not necessarily be true It will not be true always correct. So why because the example is just in front in your screen. So The thing is if if This criteria is met also it need not be a parallel gram. So hence converse of this theorem Is not all this too. I hope you understood this. Let's meet In the another session and with another theorem. Thank you