The Angle Addition Postulate

The Segment Addition Postulate has a twin for angles. When a ray falls in the interior of an angle, it splits that angle into two parts — and the parts add up to the whole. That is the Angle Addition Postulate: if $D$ is in the interior of $\angle ABC$, then $m\angle ABD + m\angle DBC = m\angle ABC$.

Just like on a line, the moment a figure becomes an equation the properties of equality take over: add the shared angle to both sides, subtract it away, or substitute equals for equals. The same reasoning that proves things about lengths proves things about angle measures.

The idea

Fan four rays $BA$, $BD$, $BE$, $BC$ out from a shared vertex $B$, in order. The Angle Addition Postulate gives $m\angle ABE = m\angle ABD + m\angle DBE$ and $m\angle DBC = m\angle DBE + m\angle EBC$. If the two outer angles are congruent, adding the shared middle angle $\angle DBE$ to both makes the whole angles equal — so $\angle ABE \cong \angle DBC$. Perpendicular rays and angle bisectors give you even shorter routes to the same kind of conclusion.

Build the proof

Read the Given and Prove, then supply the reason that justifies each statement. Click an angle in the figure to find where it appears in the proof, or click a statement to see the angle it is about. Only a step that genuinely follows from the ones above it will lock into place.