Circle 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.
A central angle has the same measure as its intercepted arc. An inscribed angle measures half its intercepted arc. A tangent to a circle is perpendicular to the radius at the point of tangency. In Measurement, Circles, and 3D Solids, students need to read the diagram, name the relationship, choose a theorem or formula, and justify why the result follows. The expanded practice now includes fluency, transfer, cumulative review, and proof-style reasoning so Geometry feels connected instead of isolated by topic.
What you'll learn
- Use central angles
- Use inscribed angles
- Recognize tangent-radius relationships
Worked example
Problem. An inscribed angle intercepts an arc of 110 degrees. Find the angle.
- An inscribed angle is half its intercepted arc.
- 110 divided by 2 is 55.
- The angle is 55 degrees.
- Connect the result back to Circle Angle Theorems so the geometric relationship is explicit.
Answer: 55 degrees
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. A central angle intercepts a 72-degree arc. What is the angle?
Show solution
- Central angle equals intercepted arc.
Answer: 72
2. An inscribed angle intercepts a 90-degree arc. What is the angle?
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- Half of 90 is 45.
Answer: 45
3. An inscribed angle is 38 degrees. What is its intercepted arc?
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- Double the inscribed angle.
Answer: 76
4. A tangent meets a radius at the circle. What angle is formed?
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- Tangent and radius are perpendicular.
Answer: 90
5. Two inscribed angles intercept the same arc. One is 64 degrees. What is the other?
Show solution
- Inscribed angles intercepting the same arc are congruent.
Answer: 64
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