Grade: Grade 10 Subject: Mathematics Unit: Geometry Theorems Lesson: 2 of 6 SAT: Geometry+Trigonometry ACT: Math

Circle Theorems

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This lesson covers the essential circle theorems you need for geometry and standardized tests. Understanding the relationships between central angles, inscribed angles, and arcs is fundamental to solving circle problems.

Key Definitions

  • Central Angle: An angle whose vertex is at the center of the circle
  • Inscribed Angle: An angle whose vertex is on the circle and whose sides contain chords of the circle
  • Arc: A portion of the circle's circumference
  • Intercepted Arc: The arc that lies in the interior of an angle and has endpoints on the angle's sides

The Inscribed Angle Theorem

The measure of an inscribed angle is half the measure of its intercepted arc (or half the central angle that subtends the same arc).

Formula: Inscribed Angle = (1/2) x Central Angle

Corollaries of the Inscribed Angle Theorem

  • Inscribed angles that intercept the same arc are congruent
  • An inscribed angle that intercepts a semicircle (diameter) is a right angle (90 degrees) - Thales' Theorem
  • Opposite angles of an inscribed quadrilateral are supplementary (sum to 180 degrees)

Examples

Example 1

Problem: A central angle measures 80 degrees. Find the inscribed angle that subtends the same arc.

Solution: Inscribed angle = (1/2) x 80 = 40 degrees

Example 2

Problem: An inscribed angle measures 55 degrees. Find the central angle.

Solution: Central angle = 2 x 55 = 110 degrees

Example 3

Problem: Two inscribed angles subtend the same arc. One measures 42 degrees. What does the other measure?

Solution: Inscribed angles subtending the same arc are congruent, so the other angle also measures 42 degrees.

Practice

Try these problems to reinforce your understanding of circle theorems.

Problem 1: A central angle is 120 degrees. Find the inscribed angle subtending the same arc.

Answer: 60 degrees

Problem 2: An inscribed angle is 65 degrees. Find the central angle subtending the same arc.

Answer: 130 degrees

Problem 3: An inscribed angle subtends a diameter. What is its measure?

Answer: 90 degrees (by Thales' Theorem)

Check Your Understanding

  1. What is the relationship between an inscribed angle and a central angle subtending the same arc?
  2. What special property does an inscribed angle have when it subtends a semicircle?
  3. If two inscribed angles subtend the same arc, what can you conclude about them?

Next Steps

  • Review the inscribed angle theorem until you can apply it confidently
  • Practice identifying central and inscribed angles in diagrams
  • Move on to Guided Practice for more complex problems