Question

In a cyclic quadrilateral two adjacent angles are 40° and ${\displaystyle \frac{{\pi }^{\text{c}}}{3}}$. Find the angles of the quadrilateral in degrees.

Answer 1

Let ABCD be the cyclic quadrilateral such that ∠A = 40° and ∠B = ${\displaystyle \frac{{\pi }^{\text{c}}}{3}}$

= ${\displaystyle {(\frac{\pi }{3}\times \frac{180}{\pi })}^{\circ } ...[\because {\theta }^{\text{c}} ={(\theta \times \frac{180}{\pi })}^{\circ }]}$

= 60°

∠A  + ∠C = 180°     ...${\displaystyle \left[\begin{array}{c}\text{Opposite angles of a cyclic}\\ \text{quadrilateral are supplementary}\end{array}\right]}$

∴ 40° + ∠C = 180°

∴ ∠C = 180° – 40° = 140°

Also, ∠B + ∠D = 180° ....${\displaystyle \left[\begin{array}{c}\text{Opposite angles of a cyclic}\\ \text{quadrilateral are supplementary}\end{array}\right]}$

∴ 60° + ∠D = 180°

∴ ∠D = 180° – 60° = 120°

∴ The angles of the quadrilateral in degrees are 40°, 60°, 140° and 120°.