Explore the angle and arc relationships for two intersecting lines that intersect a circle.

• Students will understand that if the intersection point P of two lines lies inside a circle, then the measure of the angle formed by the two secants is equal to the average of the measures of the arcs intercepted by that angle and its corresponding vertical angle.
• Students will understand that if the intersection point P of two lines lies outside a circle, then the measure of the angle formed is equal to half of the difference of the measures of the arcs intercepted by that angle and its corresponding vertical angle, whether the lines are both secants, a secant and a tangent, or two tangents.

• Secant line and chord
• Tangent line
• Central angle
• Intercepted arc

This lesson involves discovering the relationship between the measure of the angle of intersection of two lines and the measures of the intercepted arcs for several cases in which two intersecting lines intersect a circle.

• If the intersection point P of two lines lies inside a circle, then the measure of the angle formed by the two secants is equal to the average of the measures of the arcs intercepted by that angle and its corresponding vertical angle.
• If the intersection point P of two lines lies outside a circle, then the measure of the angle formed is equal to half of the difference of the measures of the arcs intercepted by that angle and its corresponding vertical angle, whether the lines are both secants, a secant and a tangent, or two tangents.