How Do Secant-Tangent and Tangent-Tangent Angles Work? A Complete Guide

• Secant-Tangent Angle: Formed when a secant line and a tangent line intersect at a point outside the circle. The angle’s measure is half the difference between the measures of the intercepted arcs.
• Tangent-Tangent Angle: Formed when two tangent lines intersect outside a circle. This angle’s measure is half the intercepted arc between the two points of tangency.

• Secant-Tangent Property: For an angle formed by a secant and a tangent intersecting outside a circle, the measure of the angle is half the difference of the intercepted arcs.
• Tangent-Tangent Property: For an angle formed by two tangent lines intersecting outside a circle, the measure of the angle is half the measure of the intercepted arc.

Examples

1. A secant and a tangent meet outside a circle, forming an angle of \( 45^\circ \). If one of the intercepted arcs by the secant is \( 150^\circ \), determine the measure of the other intercepted arc.
2. Two tangents intersect outside a circle to form an angle of \( 65^\circ \). What’s the measure of the intercepted arc between the two points of tangency?
3. Given an angle of \( 30^\circ \) formed by a secant and a tangent outside a circle, if one of the intercepted arcs is \( 80^\circ \), find the measure of the other intercepted arc.

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Answers:

1. \( 60^\circ \)
2. \( 130^\circ \)
3. \( 20^\circ \)

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