An angle formed by two chords. An inscribed angle's degree measure is half that of its intercepted arc.

Proof:

Givens: circle O, chord AB, radius AO, diameter BC.

  1. Angle BAO + angle ABC = angle AOC (by remote interior angles)
  2. AO = BO (both are radii)
  3. Triangle ABO is isosceles (definition of isosceles)
  4. Angle ABO = angle ABC (base angles of isosceles triangle are congruent)
  5. 2(angle ABC) = angle AOC (by substitution)
  6. Arc AC = angle AOC (definition central angle)
  7. 2(angle ABC) = arc AC (by substitution)
  8. Angle ABC = (arc AC)/2 (by division).

Q.E.D.

Although this is a special case of this theorem (one of the chords is a diameter), it can be used to demonstrate the theorem's veracity in other circumstances.

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