[ Inscribed Quadrilaterals (Miscellaneous).

Can a circle be circumscribed around a quadrilateral whose angles, in order, are in the ratio: a) 2:4:5:3; b) $5: 7: 8: 9 ?$

#
It is sufficient to determine whether the sums of the opposite angles of a quadrilateral are equal.

## Solution

For a circle to be circumscribed around a quadrilateral, it is necessary and sufficient that the sum of the opposite angles be equal to $180^{\circ}$. Since the sum of all four angles is $360^{\circ}$, it is sufficient to determine whether the sums of the opposite angles are equal.

## Answer

a) yes; b) no