\left.\begin{array}{l}{[\quad \text { Inscribed quadrilaterals (other). }} \\ \text { [Central angle. Arc length and circumference }]\end{array}\right]

Three consecutive angles of an inscribed quadrilateral are in the ratio 1:2:3. Find all the angles of the quadrilateral.
The sums of opposite angles of a cyclic quadrilateral are equal to each other.

## Solution

Since the sums of opposite angles of a cyclic quadrilateral are equal (each sum is $180^{\circ}$), the fourth angle accounts for 2 parts. Therefore, the angles of the quadrilateral are $\frac{1}{8}, \frac{1}{4}, \frac{3}{8}$, and $\frac{1}{4}$ of the sum of all angles, i.e., $360^{\circ}$. Thus, the required angles are $45^{\circ}, 90^{\circ}, 135^{\circ}, 90^{\circ}$.

## Answer

$45^{\circ},90^{\circ},135^{\circ},90^{\circ}$.