Lesson 10.2.3

Angles Formed by Secants & Tangents

When two secants, two tangents, or a secant and tangent meet outside a circle, the angle equals half the difference of the intercepted arcs.

Introduction

This lesson completes the angle-arc trilogy: Central = arc, Inscribed = ½ arc, Outside = ½(big arc − small arc).

Past Knowledge

Inscribed angles (10.1.4). Chord angles (10.2.2). Tangent properties (10.2.1).

Today's Goal

Apply the “half the difference” formula for exterior angles.

Future Success

Segment lengths (10.2.4–10.2.5), coordinate circles (10.3).

Key Concepts

Outside Angle Formula

When vertex is outside the circle (two secants, two tangents, or secant + tangent):

The Complete Angle-Arc Summary

Vertex Location	Formula
At center	angle = arc
On circle	angle = ½ arc
Inside circle	angle = ½(arc₁ + arc₂)
Outside circle	angle = ½(far − near)

Tangent-Chord Angle (Special Case)

When a tangent and a chord meet at the point of tangency, the angle = ½ intercepted arc (same as inscribed angle).

Worked Examples

Basic

Two Secants

Two secants from an external point intercept arcs of 120° (far) and 40° (near). Find the angle.

40°

Intermediate

Secant-Tangent

A secant and tangent from point P. Far arc = 170°, near arc = 50°. Find ∠P.

60°

Advanced

Two Tangents

Two tangents from P form a 70° angle. Find the minor and major arcs.

Let minor arc = x, major arc = 360 − x.

→

Minor = 110°, Major = 250°.

Minor = 110°, Major = 250°

Common Pitfalls

Sum vs. Difference

Inside = ½(sum). Outside = ½(difference). The most common mistake is using the wrong one. Remember: outside is always smaller, so you subtract.

Far Minus Near (Not Vice Versa)

Always subtract the smaller (nearer) arc from the larger (farther) arc. The result should be positive.

Real-Life Applications

Astronomy — Viewing Angles

When observing a planet from Earth, the angular size depends on the arcs subtended. The secant-tangent relationship helps astronomers calculate apparent sizes.

Security Cameras

A security camera outside a circular building uses the secant-tangent formula to calculate the visible arc of the building's perimeter from its mounting position.