 Hello and welcome to the session. Let us understand the following problem today. A, B, C, D is a cyclic quadrilateral. C figure 3.7, find the angles of the cyclic quadrilateral. Now let us look at the figure. This is a cyclic quadrilateral A, B, C, D with the given angles. Now, before writing the solution, we see that sum of the opposite angles of a cyclic quadrilateral is 180 degree. That is, angle A plus angle C is equal to 180 degree and angle B plus angle D is equal to 180 degree. Now this result is our key idea to the question. Now let us proceed with our solution. Given to us that, angle A is equal to 4y plus 20, angle B is equal to 3y minus 5, angle C is equal to minus 4x and angle D is equal to minus 7x plus 5. Now by the key idea, we know that angle A plus angle C is equal to 180 degree by key idea. So it implies angle A that is 4y plus 20 plus angle C that is minus 4x is equal to 180 degree. Which implies 4y plus 20 minus 4x is equal to 180. Which implies 4x minus 4y is equal to 20 minus 180. Which implies 4x minus 4y is equal to minus 160. Now taking 4 common we get x minus 5 is equal to minus 40. This is our equation 1. Now also, here angle B plus D is equal to 180 degree by key idea. Which implies angle B that is 3y minus 5 plus angle D that is minus 7x plus 5 is equal to 180 degree. Which implies 3y minus 5 minus 7x plus 5 is equal to 180 degree. Which implies 3y minus 7x this and this gets cancelled is equal to 180 degree. Which implies 7x minus 3y is equal to minus 180 degree. This is our equation 2. So we have got our 2 equations as x minus y is equal to minus 40 and 7x minus 3y is equal to minus 180. Equation 1 and equation 2. Now multiplying equation 1 by 3 we get here 3x minus 3y is equal to minus 120. And writing equation 2 as it is we get 7x minus 3y is equal to minus 180. Now subtracting them do not forget to change the sign while subtracting. We see that minus 3y plus 3y gets cancelled and we get 3x minus 7x is equal to minus 4x is equal to minus 120 plus 180 is equal to plus 60. Which implies x is equal to minus 60 by 4 which gets cancelled by 15. So which implies x is equal to minus 15. Now substituting x is equal to minus 15 in equation 1 that is x minus y is equal to minus 40 which implies minus 15 minus y is equal to minus 40. Which implies minus y is equal to minus 40 plus 15 which implies minus y is equal to minus 25 where minus and minus gets cancelled so which implies y is equal to 25. Therefore angle A is equal to 4y plus 20 which is equal to 4 multiplied by 25 plus 20 which is equal to 100 plus 20 therefore angle A is equal to 120 degree. Now angle B is equal to 3y minus 5 which is equal to 3 multiplied by 25 minus 5 which is equal to 75 minus 5 therefore angle B is equal to 70 degree. Now angle C is equal to minus 4x which implies minus 4 multiplied by minus 15 therefore angle C is equal to 60 degree. Now angle D is equal to minus 7x plus 15 plus 5 which is equal to minus 7 multiplied by minus 15 plus 5 which is equal to 105 plus 5 therefore angle D is equal to 110 degree. Hence angle A is equal to 120 degree, angle B is equal to 70 degree, angle C is equal to 60 degree and angle D is equal to 110 degree. This is our required answer. I hope you understood the problem. Bye and have a nice day.