Angles formed by lines meeting a circle

1) The angle between a tangent and a chord from the point of tangency equals half the arc lying inside the angle. 2) The angle formed by two chords that intersect inside a circle equals half the sum of the arcs subtended by that angle and its vertical angle. 3) The angle between two secants from a point outside the circle equals half the difference of the arcs between them. 4) The angle between two tangents from an outside point equals 180° minus the smaller arc between the points of tangency. To prove the first theorem, draw a chord parallel to the tangent and use alternate angles and the inscribed angle. The others follow from the exterior angle property and the inscribed angle theorem.