We know that BCD + mC = 180, since they are supplementary.
We also know that the sum of the interior angles of a triangle is 180, mA + mB + mC = 180.
Now,
BCD + mC = 180 => 180 - mC = BCD; and,
mA + mB + mC = 180 => 180 - mC = mA + mB;
It's clear to see that, from the two equations above, we have,
BCD = mA + mB.
QED.
