 Hi and welcome to the session. Let us discuss the following question. The question says if the non-palest sites of a trapezium are equal, prove that it is cyclic. Let's make a diagram to understand this question. This is the trapezium ABCD in which non-palest sites that is AD and BC are equal. We have to prove that ABCD is a cyclic trapezium. Let's first write down the given information. We are given that ABCD is a trapezium in which non-palest sites that is AD and BC are equal. Now we will construct a line BE perpendicular to AB and CF perpendicular to AB. We have drawn DE perpendicular to AB and CF perpendicular to AB and we have to prove that ABCD is a cyclic trapezium. 10.12, we know that if the sum of a pair of opposite angles of the quadrilateral is 180 degrees and the quadrilateral is cyclic. So in order to prove that ABCD is a cyclic trapezium, it is sufficient to show that angle V plus angle D is equal to 180 degrees. Let's now begin with the proof in triangle DEA and triangle CFP. AD is equal to BC. It is given to us angle DEA is equal to angle CFP because they both are of 90 degrees. And DE is equal to CF because distance between two parallel lines is always same. Angle DEA is congruent to triangle CFP by Ritch's congruence rule and hence angle A is equal to angle V and angle ADE is equal to angle BCF by CPCT. Now consider angle ADE is equal to angle BCF. Angle ADE is equal to angle BCF implies 90 degree plus angle ADE is equal to 90 degree plus angle BCF. Angle EDC is also equal to 90 degree and angle FCD is also equal to 90 degree. So 90 degree plus angle ADE is equal to 90 degree plus angle BCF implies angle EDC plus angle ADE is equal to angle FCD. Now angle EDC plus angle ADE is equal to angle ADC and angle FCD plus angle BCF is equal to angle BCD. So this implies angle D is equal to angle C. We have already proved above that angle A is equal to angle B and we have got now that angle D is equal to angle C. In trapezium ABCD angle A plus angle E plus angle C plus angle D is equal to 360 degree because sum of angles of a coordinate rule is 360 degree. Angle A is equal to angle B and angle D is equal to angle C so we can substitute angle B in place of angle A and angle D in place of angle C. So by substituting we have angle B plus angle B plus angle D plus angle D is equal to 360 degree. And this implies angle B plus angle D is equal to 180 degree. Hence we have proved that ABCD is a cyclic trapezium. This completes the session. Bye and take care.