Supplements of the same angle are equal

Depends on: Protractor axiom (angles can be manipulated as real numbers).

Statement

Let $\alpha$, $\beta$, $\gamma$ be three angles. If $\alpha$ and $\gamma$ are supplementary, and $\beta$ and $\gamma$ are also supplementary — that is,

then $\alpha = \beta$. In other words, the two supplements of the same angle must be equal.

Proof

Place the two given equations side by side — the left figure gives $\alpha + \gamma = 180^{\circ}$, the right figure gives $\beta + \gamma = 180^{\circ}$, and they share the same $\gamma$.

By Protractor axiom, angles are real numbers and can be added and subtracted. Subtracting the two equations:

$\gamma$ appears on both sides and cancels — geometrically, this is erasing the same $\gamma$ arc from both figures, after which the leftover $\alpha$ and $\beta$ arcs are automatically equal. Simplifying gives $\alpha - \beta = 0$, i.e. $\alpha = \beta$. $\blacksquare$

Immediate consequences

• Algebraization of the straight angle: for any pair that adds up to a straight angle ($180^{\circ}$), the "other half" of a given piece is unique.
• Substitution of equal angles: in any expression involving the supplement of $\gamma$, you can substitute the supplement as a single block — it doesn't matter whether you call it $\alpha$ or $\beta$.
• Symmetry across a line: the two angles formed when a ray splits one side of a line, if each is supplementary to the same external angle, must be equal to each other — this is the smallest algebraic component used later when discussing "linear pairs" and "vertical angles".

Remarks

This conclusion is the twin of Complements of the same angle are equal (the $90^{\circ}$ version), with an identical proof structure — only the constant changes from $90^{\circ}$ to $180^{\circ}$. Both rest directly on Protractor axiom: once angles are allowed to add and subtract like real numbers, "cancelling the same term" is immediately available, and geometry exits the stage.