 welcome you all again to another session on gems of geometry and continuing with our series of various theorems so we are back again with another theorem here it's it will not be an appropriate thing to call it a theorem it's basically a lemma which is a supporting argument basically but you know we can just for convenience sake sake and the continuity we are just calling it theorem here now if it says if two quads of a circle subtend different acute angles at points on the circle the smaller angle belongs to the shorter quad now the language might appear to be a little you know difficult to understand but what does this theorem say is that if you have two quads in this case if you can see there are two quads one is AB and another is CD okay now they are subtending two angles on the circle if you see one is AB is subtending an angle alpha and CD or other AB is subtending angle beta angle beta if you see angle beta is nothing but angle a PB angle a PB and CD is subtending angle alpha which is nothing but CQB CQD now what this theorem is saying is this theorem is saying that if AB is less than CD so if AB is less than CD then angle subtended by AB that is beta will also be less than angle subtended by CD that is alpha okay so if AB is less than CD if this condition is fulfilled then the angle subtended by AB that is beta will be less than angle subtended by CD which is alpha this is what we need to prove and I think we should be able to prove it by whatever knowledge we have so if you see what is beta by the way what is alpha so alpha is nothing but half of angle CD isn't it CD half of CD because angle subtended by a chord at the center is twice that of angle subtended by the same chord on the any any other point on the circle right similarly beta is equal to half of angle a OB so this is known now we know that what do we know so let's say AB is given so what is given by the way AB is shorter AB is smaller than CD now if AB is smaller than now we need to also understand that in this given case we have taken AB parallel to CD if you see AB here is shown to be parallel to CD the result will not be impacted if AB is not parallel to CD why because any other equal chord let's say if I had AB here right it would subtend the same angle beta at any other location like that so this also will be beta right so we don't really need to be bothered about the fact that how come we have taken AB parallel to CD because in any other orientation of AB also it will subtend the same angle beta so hence without the loss of generality we can take this special case right because in this case also there is no loss of the general general thing that AB if the AB length is maintained it will subtend an angle beta only at any other location it will be easier for us to prove in this way so hence we have you know I have taken that AB is parallel to CD and then what I have done is let OR be perpendicular to AB and OR be perpendicular to CD okay so this is 90 degrees here this is 90 degree here now we know that if you look carefully angle COR is definitely greater than angle AOR why because part is always less than the whole so COR is greater than AOR this implies twice angle COR is greater than twice angle AOR okay right now twice angle COR will be nothing but angle COD if you look carefully angle COD is greater than angle AOR isn't it because if you see this perpendicular by this thing the perpendicular OR will divide the triangles sorry the angle COD into two equal parts so this is established COD is greater than AOB if COD is greater than AOB that means half angle COD COD will be greater than half angle AOB okay now half angle COD is nothing but alpha so alpha is greater than beta correct alpha is greater than beta in this case now why is this coming because we had assumed angle COR is greater than AOR and this will happen only when AB is less than CD this condition COR is greater than AOR will happen only when AB is less than CD is it and from the diagram itself it is clear if AB is less than CD then AB is farther away AB is farther away farther away from center further away from O as compared to as compared to as compared to CD so AB is farther away from the center CD is closer so hence because of this particular property what will happen is angle AOR will be less than angle COR fair enough so hence friends if this condition is fulfilled this means this condition is fulfilled and if this is fulfilled ultimately alpha is greater than beta right right so hence angle subtended by larger COD will be larger than the angles subtended by the smaller COD okay