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

In the captivating realm of circle geometry, the dance between tangents and secants unveils a set of properties and angles that are both intriguing and foundational. Their interactions create angles with specific properties, vital for solving a wide array of geometric problems. Today, we'll dive into understanding the angles formed by secant-tangent and tangent-tangent combinations and their underlying principles. ## 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.

1. $$60^\circ$$
2. $$130^\circ$$
3. $$20^\circ$$

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