Adding and Subtracting Angle Measures

Angles are basically numbers with geometric meaning. So we can add and subtract them like numbers.

Right, since $\angle BAC = \angle BAD + \angle DAC$, we have that it is $20^\circ + 15^\circ = \boxed {35^\circ}$.

We could have gone backwards if we know, in advance, that $\angle DAC = 15^\circ$ and $\angle BAC = 35^\circ$ and we didn't know what $\angle BAD$ measured? We could just subtract to get $20^\circ$.