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

A secant-tangent angle formed outside a circle equals half the difference of the two intercepted arcs; a tangent-tangent angle follows the same rule. When the vertex sits on the circle, the angle is half the intercepted arc instead. Below: each case with a worked example.

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• 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.

Worked examples: secant-tangent and tangent-tangent angles

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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