In the triangle $A B C$, angle $C$ is a right angle. From the center $C$ with radius $A C$, an arc $A D E$ is described, intersecting the hypotenuse at point $D$ and the leg $C B$ at point $E$.

Find the angular measures of the arcs $A D$ and $D E$, if $\angle B=40^{\circ}$.
Triangle $A C D$ is isosceles.

## Solution

$\angle A=90^{\circ}-\angle B=50^{\circ}$, triangle $A C D$ is isosceles, $\angle A C D=180^{\circ}-2 \angle A=80^{\circ}, \angle D C B=90^{\circ}-\angle A C D=$ $10^{\circ}$.

![](https://cdn.mathpix.com/cropped/2024_05_06_7041536e9c9bacd5befeg-62.jpg?height=486&width=669&top_left_y=0&top_left_x=692)

## Answer

$80^{\circ}, 10^{\circ}$.