 Hello and welcome to the session. I am Asha and I am going to help you solve the following question which says show that diagonals of a square are equal and bisect each other at right angles. Let us now begin with the solution and here we are given a square A, B, C, D and we have to show that the diagonals are equal that is A, C is equal to B, D and they bisect at right angles that is A, O is equal to O, C and D, O is equal to O, B and all these four angles are right angles that is A, C and B, D intersect at right angles. Now consider a triangle A, D, C and triangle B, C, D here side D, C is common to C, D, angle A, D, C is equal to angle B, C, D is equal to 90 degree since angles of a square 90 degree each also A, D is equal to B, C since opposite sides of a square are equal. Now from these three conditions by right-hand side rule it follows that triangle A, D, C is concurrent to triangle B, C, D and this implies that A, C is equal to B, D since by C, P, C, D that is if two triangles are congruent then their corresponding sides are equal. So this implies that diagonals of square A, B, C, D are equal. Now we have to show that D, O is equal to O, B and A, O is equal to O, C and they intersect at right angles. Now in triangle A, O, B and triangle D, O, C, AB is equal to DC since sides of a square are equal and also since AB is parallel to DC and BD is any transversal so angle A, B, D is equal to angle B, DC that is we can say angle AB, O is equal to angle O, D, C since AB is parallel to DC and DB is the transversal this implies angle ABD is equal to angle B, DC since alternate interior angles are equal. Similarly we can say that angle BAO is equal to angle TO since again these two lines are parallel and now AC is the transversal this implies angle ABD is equal to angle CDB. Now from these three conditions by ASA exam we can say that triangle ABO is congruent to triangle DOC which further implies that AO is equal to OC and DO is equal to OB so this is by CPCT according to which if two triangles are congruent then their corresponding parts are also equal. So with the help of ASA exam we have shown that the two diagonals bisect each other and now it is considered triangles AO, D and triangle AO, B. Now in these two triangles AO is equal to OA this is common sight DO is equal to OB just now we have proved AD is equal to AB since sites of a square are equal. So by ASA exam triangle AOD is congruent to triangle AOB so this implies angle AOD is equal to angle AOB since again by CPCT. Now angle AOD plus angle AOB is equal to 180 degree and since these two angles are equal it follows that two times angle AOD is equal to 180 degree which further implies angle AOD is equal to 90 degree. Similarly we can show that angle AOB is equal to angle BOC is equal to angle DOC is equal to 90 degree this implies that diagonals of a square are equal bisect at right angles. So this completes the session take care and bye for now.