Circles and Angles

Central Angle

• Definition: Vertex is at the center of the circle.
• Example: In circle C with points A and B, if ∠ACB = 50 degrees, then the measure of arc AB is also 50 degrees.

Inscribed Angle

• Definition: Vertex is on the circle.
• Example: ∠ABC is an inscribed angle with chords AB and BC. If ∠ABC = 30 degrees, the measure of arc AC is 60 degrees (double the angle).

Tangent-Chord Angle

• Definition: Formed when a tangent segment meets a chord.
• Example: ∠ABC is 25 degrees; the measure of intercepted arc AB is 50 degrees (double the angle).

Chord-Chord Angle

• Definition: Formed by the intersection of two chords.
• Calculation: Average of the intercepted arcs.
• Example: If arc AC = 100 degrees and arc DE = 60 degrees, then ∠ABC = 80 degrees (average of the two arcs).
• Problems:
  • If arc DE = 70 degrees and ∠DBE = 55 degrees, then arc AC = 40 degrees.
  • If arc DE = 110 degrees, arc AC = 50 degrees, then ∠EBC = 100 degrees.
  • If ∠ABD = 115 degrees and arc AC = 75 degrees, then arc CE = 125 degrees (by calculating the missing arc).

Secant-Secant Angle

• Definition: Formed by two secant segments with a common endpoint.
• Calculation: One half the difference of the intercepted arcs.
• Example: If arc AC = 110 degrees and arc DE = 60 degrees, then ∠B = 25 degrees.

Secant-Tangent Angle

• Definition: Formed by a secant segment and a tangent segment.
• Example: If arc AC = 130 degrees and ∠B = 30 degrees, then arc DC = 70 degrees.

Tangent-Tangent Angle

• Definition: Formed by two tangent segments.
• Calculation: One half the difference of the major and minor arcs.
• Example: If the major arc AXC = 220 degrees, then the measure of angle B is 40 degrees.

Central Angle and Inscribed Angle Problem

• Example: ∠BDC = 40 degrees, so arc BC = 40 degrees. Then ∠BAC (inscribed angle) = 20 degrees (half of arc BC).

Comprehensive Example Problem

• Given: arc AC = 9x + 18, arc DE = 5x + 10, and ∠ABC = x² + 6.
• Steps:
  • Combine and solve equations.
  • X-value solutions: 8 or -1.
  • Using x = 8, arc AC = 90 degrees.

Closing Problems

• Use various rules to solve for unknown angles and arcs based on given values.

Key Forms and Formulas

• Inscribed Angle: Half the measure of the intercepted arc.
• Central Angle: Equal to the intercepted arc.
• Tangent-Chord Angle: Twice the measure of the angle.
• Chord-Chord Angle: One half the sum of intercepted arcs.