 Hello and welcome to the session. I am Asha and I am going to help you solve the following question which says, Diagonals AC of a parallelogram, ABCD, bisects angle A. Show that first, if bisects angle C also, second, ABCD is a rhombus. So let us begin with the solution and we are given in parallelogram, ABCD angle DAC is equal to angle BAC and we have to show first that angle DCA is equal to angle BCA and second that ABCD is a rhombus. So let us consider angle ADC and triangle ABC. Now in these two triangles, side AC is common to CA since both the triangles and DC is equal to AB since opposite sides of a parallelogram are equal and also AB is parallel to DC since opposite sides of a parallelogram are equal and AC is any transversal. So this implies angle BAC is equal to angle DCA since if two parallel lines are intersected by a transversal then alternate interior angles are equal. Now from this, this and this point we find that by side angle side criteria triangle ADC is congruent to triangle ABC. Now these two triangles are congruent this implies angle DCA is equal to angle BCA since by CPCT since if two triangles are congruent then their corresponding parts are also equal. So this implies AC by six angle C also. Now we will show the second part that ABCD is a rhombus. Now consider triangles ADC and triangle ABC angle DAC is equal to angle BAC this is given to us since the diagonal AC by six angle A side AC is common to CA in the first part we are shown that AC by six angle C therefore we have angle DCA equal to angle BCA since by first part. So by angle side angle property we find that triangle ADC is congruent to triangle ABC this implies AD is equal to side AB since by CPCT that is if two triangles are congruent then their corresponding parts are equally and we are given that ABCD is a parallelogram. So this implies side AD is equal to side BC and from this equation and from this equation we find that AD is equal to AB is equal to BC and since BC is equal to AD since they are opposite sides of a parallelogram AB is also equal to DC. So all these four are equal and this implies that parallelogram ABCD is a rhombus hence we have proved that AC by six angle C parallelogram ABCD is a rhombus. This completes the session hope you enjoyed it take care and have a good day.