the angle addition postulate
If you open a fan partway, then open it the rest of the way, the total spread is the sum of the two openings. The angle addition postulate makes that obvious idea exact for angles that share a vertex and a side.
The postulate states: if point D lies in the interior of angle ABC (so ray BD is inside the angle), then m(angle ABD) + m(angle DBC) = m(angle ABC). The two smaller adjacent angles, side by side, add up to the big one they make together. Read backwards, it lets you split a known angle: m(angle DBC) = m(angle ABC) - m(angle ABD). The key requirement is that ray BD genuinely lies between the two outer sides; if it pointed outside the angle, the sum would not hold.
This is the angle-world twin of the segment addition postulate, and it is the tool behind angle bisectors and most angle-chasing proofs. Because an angle bisector splits an angle into two equal halves, the postulate immediately gives m(angle ABD) = m(angle DBC) = (1/2)·m(angle ABC). Combined with the angle-pair relationships (complementary, supplementary, linear pairs), it turns the geometry of angles into solvable algebra inside a two-column proof.
Ray BD lies inside angle ABC with m(angle ABD) = 28° and m(angle DBC) = 47°. Then m(angle ABC) = 28 + 47 = 75°. If instead the whole angle ABC = 90° and m(angle ABD) = 35°, then m(angle DBC) = 90 - 35 = 55°.
An interior ray splits an angle into two parts that sum to the whole.
Ray BD must lie in the interior of angle ABC for the sum to work; if it falls outside, m(angle ABD) + m(angle DBC) no longer equals m(angle ABC).