 Hi, and how are you all today? My name is Priyanka and the question says, in figure 9.25, diners AC and BD of quadrilateral intersect at O such that OB is equal to OT. If AB is equal to AC, show that area of DOC is equal to AOB, area of DCB is equal to area of ACB and thirdly DA is parallel to CB or ABCD is a parallelogram. Now this is the figure that we need to refer. Here we are given that this is point O where the diners are intersecting each other and diners are AC and BD and they are intersecting each other such that OB is equal to OT. Also AB is equal to CD. Now we need to prove that area of DOC is equal to area of AOB then area of DCB this whole triangle is equal to ACB and then DA is parallel to CB or this whole is a parallelogram. Now I think you must have got some of the idea while I was explaining the question itself. First of all we will write whatever is given to us. We are given that AB CD is a quadrilateral whereas OB is equal to OT and AB is equal to CD. We need to prove three things. First, area of DOC is equal to area of AOC, AOB. Second, area of DCB is equal to area of ACB and thirdly DA is parallel to CB or ABCD is a parallelogram. Then I start with our proof of the first part that is area of DOC is equal to area of AOB. We know that the area of two triangles are equal when they are congruent. So let us prove these two triangles as congruent triangle and then we can easily prove that they are equal in area also. So in triangle DOC and AOB we know that OB here OD is equal to OB that is given to us in the question. Angle DOC is equal to angle AOB. They are forming vertically opposite angles. E is equal to DC so we can write DC is equal to AB. This is also given to us in the question. So therefore triangle DOC is congruent to triangle AOB by SAS congruency rule. We can say that therefore area of DOC is equal to area of AOB giving the simple reason that congruent triangles in area. So this completes the first part of this question. Let us proceed on to the second part. Here we need to prove that area of DCB is equal to area of ACB. Now we know that area of DOC is equal to area of AOB. This is proved in first part OCB as you see that this is the common region of these two triangles. So if we add the area of OCB to DOC we will have this whole big triangle. So on adding area of OCB on both sides we have area of DOC plus area of OCB equal to area of AOB plus area of OCB. So area of OCB has been added to these two areas. So area of DOC plus area of OCB becomes area of DCB. So that's area of B. So this completes the second part of our proof. Let us proceed on with our third part to CB or is a parallelogram. From first part equal to area of ACB. This is what we have proved in the second part of this question. These two triangles are on the same base and their areas are equal that means they are between the same parallel lines. So we can say that therefore if on the same base then they are between the same parallel lines. So that means they are parallel to each other. Now also EA will be equal to DB by CPCT because these two triangles were equal to each other. Otherwise we can say that since TA is parallel to CB is equal to CD. So that means they will also be equal to each other and AB will be parallel to CB. Therefore by all these criteria we can say that therefore ABCD is a parallelogram. Right? So this completes the question that was given to us. Hope you enjoyed proving all the three parts and take care.