Tangent, Secant, and Chord Angle Theorems
A free Geometry lesson from the “Measurement, Circles, and 3D Solids” unit, with a worked example and practice problems including step-by-step solutions.
Inscribed angles equal half the intercepted arc. Two chords meeting INSIDE a circle form an angle equal to half the SUM of the two intercepted arcs. Two secants (or a tangent + secant, or two tangents) meeting OUTSIDE a circle form an angle equal to half the DIFFERENCE of the two intercepted arcs. The power of a point gives equal products: chord-chord AE * EB = CE * ED; tangent-secant tangent^2 = whole-secant * near-piece.
What you'll learn
- Apply the inscribed-angle theorem (inscribed angle is half the intercepted arc)
- Apply tangent-chord, two-chord, two-secant, tangent-secant angle relationships
- Use the intersecting-chord and tangent-secant power-of-a-point segment rules
Worked example
Problem. Two chords AB and CD intersect inside a circle at E. AE = 2, EB = 6, CE = 3. Find ED.
- Intersecting chords: AE * EB = CE * ED.
- 2 * 6 = 3 * ED, so 12 = 3 * ED.
- ED = 4.
- Connect the result back to Tangent, Secant, and Chord Angle Theorems so the geometric relationship is explicit.
Answer: 4
Lesson preview: this page shows 5 practice items. The interactive activity contains 28 practice items and 15 quiz items (typed response and multiple choice). Counts include repeated prompts and variants.
Practice problems
1. An inscribed angle intercepts an arc of 100 degrees. Find the angle in degrees.
Show solution
- Inscribed angle = arc / 2.
- 100 / 2 = 50.
Answer: 50
2. A central angle intercepts an arc of 80 degrees. Find the central angle.
Show solution
- A central angle equals its intercepted arc.
Answer: 80
3. A tangent-chord angle intercepts an arc of 80 degrees. Find the angle.
Show solution
- Tangent-chord angle = arc / 2.
- 80 / 2 = 40.
Answer: 40
4. Two chords inside a circle intercept arcs of 60 and 100 degrees. Find the angle they form.
Show solution
- Inside the circle: angle = (60 + 100) / 2.
- = 160 / 2 = 80.
Answer: 80
5. Two secants from an external point intercept arcs of 120 and 40. Find the angle.
Show solution
- Outside the circle: angle = (120 - 40) / 2.
- = 80 / 2 = 40.
Answer: 40
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