3 [ Angles subtended by equal arcs and equal chords]

The vertex of an angle measuring $70^{\circ}$ is the origin of a ray that forms angles of $30^{\circ}$ and $40^{\circ}$ with its sides. Perpendiculars are dropped from a point $M$ on this ray to the sides of the angle, with the feet of the perpendiculars being $A, B$, and $C$. Find the angles of triangle $A B C$.

#
Let $M A$ and $M C$ be the perpendiculars dropped from a point on the sides of a given angle with vertex $O$, and $M B$ be the perpendicular dropped onto the ray between the sides of angle $A O C$, where $\angle A O B=30^{\circ}, \angle C O B=40^{\circ}$.

Points $A, B$, and $C$ lie on a circle with diameter $O M$, so $\angle A C B=\angle A O B=30^{\circ}$. Similarly, $\angle B A C=$ $\angle C O B=40^{\circ}$.

![](https://cdn.mathpix.com/cropped/2024_05_06_0ef0b80dacb6d98eff5ag-30.jpg?height=617&width=640&top_left_y=0&top_left_x=709)

## Answer

$30^{\circ}, 40^{\circ}, 110^{\circ}$.