How did you include that diagram? That worked awesome! I want to be able to do that!
The key to solving this problem is to consider the meaning of supplemental angles. They are pairs of angles that, when laid side to side, form a straight line with their outer rays. That's an awkward way of saying it. Mathematically, they are two angles whose measures add to 180 degrees, which is the measure of a straight line.
m∠ABD + m∠CBD = 180 Definition of supplementary angles
m∠BDE + m∠CDB = 180 Definition of supplementary angles
m∠CBD = 180 – m∠ABD Algebraic manipulation of above equation
m∠CDB = 180 – m∠BDE Algebraic manipulation of above equation
m∠CBD = 180 – m∠BDE Substitution of equivalents
m∠CBD = m∠CDB Substitution of equivalents