Circle theorems

• What does the alternate segment theorem state?

  The angle between a tangent and a chord equals the angle in the alternate segment.
• How does the angles in the same segment theorem relate to angles in a circle?

  Angles in the same segment are equal.
• What does the angle at the center theorem state?

  The angle at the center is twice the angle at the circumference.
• What is the measure of the angle in a semicircle?

  90 degrees
• What happens to a chord when a perpendicular is drawn from the center of a circle?

  The perpendicular bisects the chord into two equal parts.
• What is the sum of the opposite angles in a cyclic quadrilateral?

  180 degrees
• What is the relationship between a tangent and a radius at the point of contact?

  The angle between a tangent and radius is 90 degrees.
• What can be said about tangents that meet at the same point on a circle?

  Tangents which meet at the same point are equal in length.
• What are the key circle theorems related to angles?

  • Alternate segment theorem: Angle between tangent and chord equals angle in alternate segment.
  • Angles in the same segment theorem: Angles in the same segment are equal.
  • Angle at the center theorem: Angle at center is twice the angle at circumference.
  • Angles in a semicircle: Angle in a semicircle is 90 degrees.
• What are the properties of chords and cyclic quadrilaterals in circles?

  • Perpendicular from center bisects the chord.
  • Opposite angles in a cyclic quadrilateral total 180 degrees.
• What are the properties of tangents in relation to circles?

  • Angle between tangent and radius is 90 degrees.
  • Tangents meeting at the same point are equal in length.