High School Mathematics - 2 8.6 Circles - Theorems

Statement: If two arcs are congruent, then their chords are equal.
Given: Arc ASB = Arc CTD
To prove: AB = CD

Proof: OA = O^{l}C, OB = O^{l}D (radii)
Angle AOB = Angle CO^{l}D (given that the arcs are congruent)
Hence by SAS Postulate on congruence AOB is congruent to CO^{l}D
Hence AB = CD

Statement: There is only one circle passing through three non collinear points.

Example: In the given figure prove that AB = CD

Solution: Arc BAD congruent Arc CDA
Hence Arc BAD - Arc AD = Arc CAD - Arc AD
Arc BA is congruent Arc CD
Hence AB = CD