Question

ABCD is a cyclic quadrilateral such that, $\angle A = {(4y + 20)^\circ},\angle B = {(3y - 5)^\circ},\angle C = {( - 4x)^\circ}$ and $\angle D = {( - 7x + 5)^\circ}$ . Find the four angles.

Answer 1

Hint: Here we go through the property of cyclic quadrilateral as we know that the opposite angles of a cyclic quadrilateral are supplementary and we apply the properties to form the equation.

Complete step-by-step answer:
Here in the question it is given that ABCD is a cyclic quadrilateral in which the angles are given as,
$\angle A = {(4y + 20)^\circ},\angle B = {(3y - 5)^\circ},\angle C = {( - 4x)^\circ}$ and $\angle D = {( - 7x + 5)^\circ}$
Now as we know the sum of opposite angles in cyclic quadrilateral are supplementary so we can say that,
$\angle A + \angle C = {180^\circ}$ [Sum of angles in cyclic quadrilateral are supplementary]
$
   \Rightarrow 4y + 20 - 4x = 180 \\
   \Rightarrow 4y - 4x = 160 \\
 $
$\therefore y - x = 40$ …………….. Let it be as equation (1).
Now similarly,
$\angle B + \angle D = {180^\circ}$ [Sum of angles in cyclic quadrilateral are supplementary]
$
   \Rightarrow 3y - 5 - 7x + 5 = 180 \\
   \Rightarrow 3y - 7x = 180 \\
$
$\therefore 3y - 7x = 180$ …………….. Let it be as equation (2).
Now multiply by 3 in equation (1) we get,
$ \Rightarrow 3y - 3x = 120$…… let it be equation (3).
Subtracting equation (3) from equation (2) we get,
$
   \Rightarrow - 4x = 60 \\
  \therefore x = - 15 \\
 $
Substituting value of x in equation (1) we get,
$
   \Rightarrow y = 40 + x \\
   \Rightarrow y = 40 - 15 \\
  \therefore y = 25 \\
 $
Now put the values of x and y in the angles to get the required angles.
$
   \Rightarrow \angle A = {(4y + 20)^\circ} = (4 \times 25 + 20) = {120^\circ} \\
   \Rightarrow \angle B = {(3y - 5)^\circ} = (3 \times 25 - 5) = {70^\circ} \\
   \Rightarrow \angle C = {( - 4x)^\circ} = ( - 4 \times ( - 15)) = {60^\circ} \\
   \Rightarrow \angle D = {( - 7x + 5)^\circ} = ( - 7 \times ( - 15) + 5) = {110^\circ} \\
 $

Note: Whenever we face such questions the key concept for solving the question is to first try to make the diagrams and apply the properties of the given figure and we also know that two variables are solved from two equations. So by the property of the figure try to make two equations to find out the variables.