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

Step-by-step Guide: Secant-Tangent and Tangent-Tangent Angles

Definition:

• 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.

Properties of Secant-Tangent and Tangent-Tangent Angles:

• 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

Example 1:
A secant \( AB \) and a tangent \( AC \) intersect outside a circle at point \( A \). If the intercepted arcs for the secant are \( 60^\circ \) and \( 160^\circ \), find the measure of angle \( CAB \).

Solution:
Using the secant-tangent property:
\( \text{Angle} = \frac{\text{difference of intercepted arcs}}{2} \)
\( \angle CAB = \frac{160^\circ – 60^\circ}{2} = \frac{100^\circ}{2} = 50^\circ \)

Example 2:
Two tangents \( AB \) and \( AC \) intersect outside a circle at point \( A \). The intercepted arc between the points of tangency \( B \) and \( C \) is \( 140^\circ \). Determine the measure of angle \( BAC \).

Solution:
Using the tangent-tangent property:
\( \text{Angle} = \frac{\text{intercepted arc}}{2} \)
\( \angle BAC = \frac{140^\circ}{2} = 70^\circ \)

Practice Questions:

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.

Answers:

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