Circle Theorems Lecture Notes

Key Points of a Circle

• Chord: A straight line from one side of the circle to the other.
  • Splits a circle into two segments:
    • Upper segment
    • Lower segment

Theorems

1. Angles in the Same Segment

• Theorem: Angles in the same segment are equal.
  • If angles are drawn using the same chord and within the same segment, they are equal.
  • Example: Purple, green, and blue angles are equal if they are in the same segment.
  • Often visualized without the chord; resembles a bow tie.

2. Angle in a Semicircle

• Theorem: The angle in a semicircle is 90 degrees.
  • If a chord is a diameter, angles in the same segment are right angles.

3. Angle at the Center

• Theorem: The angle at the center is twice the angle at the circumference.
  • Example: If the angle at the circumference is 50 degrees, the angle at the center is 100 degrees.

4. Cyclic Quadrilateral

• Definition: A quadrilateral where all four sides touch the circumference.
  • Opposite angles in a cyclic quadrilateral add to 180 degrees.
  • Example: If one angle is 86 degrees, the opposite must be 94 degrees.

5. Tangent and Radius

• Theorem: A tangent meets a radius at 90 degrees.
  • Tangents from the same point are equal in length.
  • The line from the center to the point of tangency bisects the angle at that point.

6. Alternate Segment Theorem

• Theorem: The angle between the tangent and chord is equal to the angle in the alternate segment.
  • Example: The angle made by the tangent and chord equals the opposite angle in the circle.

Problem-Solving Using Theorems

Example: Find the Angle ABD

• Identify angles on a straight line or in the same segment to find missing angles.
• Use theorem: Angles in the same segment are equal to find angles.

Example: Find Angle BCF

• Use isosceles triangle properties to determine equal angles.
• Use cyclic quadrilateral properties to find opposite angles summing to 180 degrees.
• Utilize the semicircle theorem for right angles.

Example: Tangent and Radius

• Use theorem: A tangent meets a radius at a right angle to find angles in triangles.
• Apply theorem: Angle at center is twice the angle at circumference to solve problems.

Additional Tips:

• Mark known angles directly on diagrams.
• Show working out clearly in problem-solving.
• Note: There can be multiple approaches to solving circle theorem problems.