A tangent is drawn through the end of a chord equal to the radius of the circle, find the angles between the tangent and the chord.

From point O, the center of the circle, draw the radii OA and OB.

By condition, the length of the chord is equal to the radius of the circle, then the triangle AOB is equilateral, and therefore all of its internal angles are 60.

By the property of the tangent, the radius drawn to the point of tangency is perpendicular to this tangent, then the angle CAO = DAO = 90. Then the angle ВAD = DAO – OAD = 90 – 60 = 300, angle CAB = CAO + OAВ = 90 + 60 = 150.

Answer: The angles between the tangent and the chord are 30 and 150.