Given: Circle O with diameter AB, arc AM = arc DM

Prove: △AMB ≅ △CMB

Given
∠ABM = ∠CBM
Substitution
△AMB ≅ △CMB
∠ABM = (1/2) * arc AM and ∠CBM = (1/2) * arc DM
ASA (Angle-Side-Angle)
Substitution
Inscribed angle measures half its intercepted arc
The angles in a linear pair are supplementary
∠CMB = 90
∠AMB = 90
Reflexive
Inscribed angle measures half its intercepted arc
arc AB = 180
Circle O with diameter AB, arc AM = arc DM
MB = MB
∠AMB = (1/2) * arc AB
Definition