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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.

What you'll learn

Why it matters: Engineers and designers use circle angle relationships in wheels, round windows, lenses, and tangent supports.

Worked example

Problem. An inscribed angle intercepts an arc of 110 degrees. Find the angle.

  1. An inscribed angle is half its intercepted arc.
  2. 110 divided by 2 is 55.
  3. The angle is 55 degrees.

Answer: 55 degrees

Practice problems

1. A central angle intercepts a 72-degree arc. What is the angle?

Show solution
  1. Warm-up: First identify exactly what the question is asking: A central angle intercepts a 72-degree arc. What is the angle?
  2. For intercepts, remember that an x-intercept has y = 0 and a y-intercept has x = 0.
  3. Central angle equals intercepted arc.
  4. Check the result by substituting or estimating: the response should match 72 and make sense in the original problem.

Answer: 72

2. An inscribed angle intercepts a 90-degree arc. What is the angle?

Show solution
  1. Warm-up: First identify exactly what the question is asking: An inscribed angle intercepts a 90-degree arc. What is the angle?
  2. For intercepts, remember that an x-intercept has y = 0 and a y-intercept has x = 0.
  3. Half of 90 is 45.
  4. Check the result by substituting or estimating: the response should match 45 and make sense in the original problem.

Answer: 45

3. An inscribed angle is 38 degrees. What is its intercepted arc?

Show solution
  1. Core Practice: First identify exactly what the question is asking: An inscribed angle is 38 degrees. What is its intercepted arc?
  2. For intercepts, remember that an x-intercept has y = 0 and a y-intercept has x = 0.
  3. Double the inscribed angle.
  4. Check the result by substituting or estimating: the response should match 76 and make sense in the original problem.

Answer: 76

4. A tangent meets a radius at the circle. What angle is formed?

Show solution
  1. Core Practice: First identify exactly what the question is asking: A tangent meets a radius at the circle. What angle is formed?
  2. Use the relevant geometric relationship first, then set up an equation from the angle measures or side relationships.
  3. Tangent and radius are perpendicular.
  4. Check the result by substituting or estimating: the response should match 90 and make sense in the original problem.

Answer: 90

5. Two inscribed angles intercept the same arc. One is 64 degrees. What is the other?

Show solution
  1. Challenge: First identify exactly what the question is asking: Two inscribed angles intercept the same arc. One is 64 degrees. What is the other?
  2. For intercepts, remember that an x-intercept has y = 0 and a y-intercept has x = 0.
  3. Inscribed angles intercepting the same arc are congruent.
  4. Check the result by substituting or estimating: the response should match 64 and make sense in the original problem.

Answer: 64

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