The angular magnitude of the arc is $110^{\circ}$. Find the angle between the chord and the extension of the radius drawn to the end of the arc.

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Let $O$ be the center of the circle, $AB$ be a chord. The required angle is adjacent to the base angle of an isosceles triangle with a vertex angle of $110^{\circ}$.

![](https://cdn.mathpix.com/cropped/2024_05_06_7041536e9c9bacd5befeg-59.jpg?height=360&width=349&top_left_y=2410&top_left_x=863)

## Answer

## [ Inscribed circle in an angle $\quad]$ Problem 52601 Topics: [Properties and criteria of isosceles triangles.] [ Sum of interior and exterior angles of a polygon ]

Tangents are drawn from the ends of an arc of $200^{\circ}$ until they intersect. Find the angle between them.

## Hint

Draw radii to the points of tangency.

## Solution

Let $A$ and $B$ be the points of tangency, $O$ be the center of the given circle, and $M$ be the point of intersection of the tangents. Then $\angle AOB = 360^{\circ} - 200^{\circ} = 160^{\circ}$.

Since $OA \perp AM$ and $OB \perp BM$, then $\angle AMB = 180^{\circ} - \angle AOB = 180^{\circ} - 160^{\circ} = 20^{\circ}$.

## Answer

$20^{\circ}$.