Tangent AN is drawn through point A to the circle with center O. Find the angles of the triangle

Tangent AN is drawn through point A to the circle with center O. Find the angles of the triangle AON if the angle AON is five times greater than the angle ANO.

Let’s draw the radius OA to the point of tangency.

Since AH is tangent, and the radius OA is drawn to the point of tangency, then by the property of the tangent, the radius of AO is perpendicular to the tangent AH, and therefore the triangle AON is rectangular.

Let the value of the angle ANO be equal to X0, then, by condition, the value of the angle AON will be equal to 5 * X0.

The sum of the interior angles of the triangle is 180, then (90 + X + 5 * X) = 18.

6 * X = 90.

X = 90/6 = 15.

Angle ANO = 150, then angle AON = 5 * 15 = 75.

Answer: The angles of the AON triangle are 15, 75, 90.