Circles

I haven't really learned this stuff at school, but I can assume that if AB and CD are parallel and chord AC is 100, then chord BD must also be 100, so that makes the measure of angle BOD = 100°

Answer: 100°

Connect AC ,  AO

Angle CAB  = 90  = angle ACB

So arc CDB =  180 = arc DCA

Then BC and AD are  diameters

Angle ABC = (1/2) 100  =  50 = Angle ADC

ABODC   is a  pentagon....the sum of  its interior angles  = 540

So  angle 

BOD =  540  - 2(90) - 2(50) =   540  - 180 - 100  =   260

So angle BOD = 360  - 260  =  100   ( just  as DS2011 found !!!