Other Angle Relationship in Circles

A polygon is inscribed in a circle if its vertices are points on the circle and its sides are chords of the circle. Equivalently, the circle is said to be circumscribed about the polygon. A polygon is circumscribed about a circle if all sides of the polygon are segments tangent to the circle; also the circle is said to be inscribed in the polygon. There are many kinds of polygons..... each with angles and sides. When we talk about the angles of a polygon, we usually mean the interior angles. A polygon has as many interior angles as sides. An equilateral triangle has three equal 60 degree angles. The sum of the angles of this and any triangle is 180 degrees.The sum of the four interior angles of a square is 360 degrees, which is the same for any quadrilateral. The sum of the interior angles increases by 180 degrees for each additional side ... pentagon, hexagon, heptagon, octagon ... as the polygons have more sides, the interior angles become larger and there are more of them, so the sum of the interior angles increases. There are other angles related to polygons: the exterior angles. An exterior angle is formed by extending one side at each vertex. As the number of sides increases, the exterior angles become smaller.

Central Angle Postulate In a circle, the degree measure of a central angle is equal to the degree measure of its intercepted arc. When two chords intersect inside a circle, four angles are formed. If two chords intersect in the interior of a circle, then the measure of each angle is one half of the measures of the arcs intercepted by the angle and its vertical angle.

If a tangent and a chord intersect at a point on a circle, then the measure of each angle formed is one half the measure of its intercepted arc.

If two chords intersect in the interior of a circle, then the measure of each angle is one half the sum of the measures of the arcs intercepted by the angle and its vertical angle.

If a tangent and a secant, two tangents, or two secants intersect in the exterior of a circle, then the measure of the angle formed is one half the difference of the measures of the intercepted arcs.

If a tangent and a secant, two tangents, or two secants intersect in the exterior of a circle, then the measure of the angle formed is one half the difference of the measures of the intercepted arcs. If a tangent and chord intersect at a point on a circle, then the measure of each angle formed is one half the measure of its intercepted arc.