Angles in a Circle

Key Concepts and Terminologies

• Center: Point inside the circle, often denoted as C or O.
• Radius (R): Distance from the center to any point on the circle. Constant for all points on the circle.
• Diameter: Line passing through the center and touching two points on the circle. Length is 2R.
• Arc: A portion of the circumference. The entire circumference is the largest arc.
• Sector: A 'slice' of the circle formed by two radii and the arc between them.
• Chord: A straight line connecting two points on the circle, but not passing through the center.
• Arc Length: The distance measured along the arc, can be found by 'straightening out' the arc and measuring it.

Types of Angles in a Circle

Central Angles

• Definition: An angle with its vertex at the center of the circle. The arms are radii.
• Properties:
  • Can be associated with a specific arc.
  • The size of a central angle is equal to the size of the arc it subtends.
  • When measuring through the diameter, the central angle is 180°.

Inscribed Angles

• Definition: An angle with its vertex on the circle itself. The arms are chords.
• Properties:
  • Multiple inscribed angles subtending the same arc have the same measure.
  • Inscribed angle is half the measure of the central angle subtending the same arc.
  • An inscribed angle opposite a diameter is always 90°.

Relationships Between Angles

• Central Angle_Size (α)

  • Central angle is subtended by an arc between its arms.
  • Two central angles sharing the same arc segment are supplementary (adding up to 360° total).

• Inscribed Angle_Size (β)

  • Inscribed angle is subtended by an arc between its arms.
  • Inscribed angles sharing the same arc segment are equal.
  • If an inscribed angle and a central angle subtend the same arc, the central angle is twice the size of the inscribed angle.

Special Cases and Theorems

• Relationship Between Central and Inscribed Angles:

  • If a central angle and an inscribed angle subtend the same arc, then the central angle is exactly 2× the inscribed angle.
  • Proof involves the properties of isosceles triangles.

• Two Types of Angles with the Same Arc:

  1. Central Angle = 2× Inscribed Angle
  2. Inscribed Angles that subtend the same arc are equal.

• Inscribed Angle Opposite the Diameter:

  • Always measures 90°.
  • Proof: The central angle subtending the arc of a diameter is 180°, meaning any inscribed angle subtending it would be 90°.

• Cyclic Quadrilateral:

  • A quadrilateral where all vertices lie on the circle.
  • Opposite angles in any cyclic quadrilateral are supplementary, i.e., α + β = 180°.

• Practice Problems Summary:

  • Use relationships between central and inscribed angles to find unknown angles.
  • Recognize inscribed angle patterns and properties.
  • Identify and work with cyclic quadrilaterals.

Important Facts

• Central angle around diameter: 180° always.
• Any inscribed angle around diameter: 90° always.
• Any inscribed angles subtending the same arc: Equal.
• Central angle vs. inscribed angle subtending the same arc: Central = 2× Inscribed.