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.
- Angle BAO + angle ABC = angle AOC (by remote interior angles)
- AO = BO (both are radii)
- Triangle ABO is isosceles (definition of isosceles)
- Angle ABO = angle ABC (base angles of isosceles triangle are congruent)
- 2(angle ABC) = angle AOC (by substitution)
- Arc AC = angle AOC (definition central angle)
- 2(angle ABC) = arc AC (by substitution)
- 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.