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A rhombus and an isosceles trapezoid are described around the same circle and have the same area. Find their acute angles.

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Let the radius of the circle be $r$, the side of the rhombus be $a$, its acute angle be $\alpha$, the lateral side of the trapezoid be $b$, and its acute angle be $\beta$. Then

$$
S_{\text {rhombus }}=a \cdot 2 r=\frac{2 r}{\sin \alpha} \cdot 2 r=\frac{4 r^{2}}{\sin \alpha}
$$

and since the semi-sum of the bases of an isosceles trapezoid circumscribed around a circle is equal to the lateral side, then

$$
S_{\text {trapezoid }}=b \cdot 2 r=\frac{2 r}{\sin \beta} \cdot 2 r=\frac{4 r^{2}}{\sin \beta}
$$

From the equality of the areas of the rhombus and the trapezoid, it follows that the sines of the angles $\alpha$ and $\beta$ are equal, and since these angles are acute, $\alpha=\beta$.

## Answer

The acute angles of the rhombus and the trapezoid are equal.

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