Supplementary Angles
Supplementary angles are two angles whose measures add up to exactly 180 degrees.
Formula
\text{angle } A + \text{angle } B = 180^\circ
Definition
Two angles are supplementary if they add up to $180^\circ$ ($\alpha + \beta = \pi$), a straight line, like two pieces that together form a straight angle; adjacent supplementary angles form a linear pair, and co-interior angles formed by a transversal crossing parallel lines are always supplementary. The identities $\sin(\pi - x) = \sin x$ and $\cos(\pi - x) = -\cos x$ express supplementarity analytically, and in a cyclic quadrilateral (inscribed in a circle), opposite angles are always supplementary, a consequence of the inscribed angle theorem.
Example
A $110^\circ$ angle and a $70^\circ$ angle are supplementary ($110+70=180$); if you cut a straight line with another line, the two angles on one side are always supplementary. When parallel lines are cut by a transversal, co-interior angles are supplementary, if one is $115^\circ$, the other is $65^\circ$. In cyclic quadrilateral $ABCD$: angle $A$ + angle $C = 180^\circ$ and angle $B$ + angle $D = 180^\circ$, a property that characterizes cyclic quadrilaterals.
Key Insight
A memory trick: S in Supplementary goes with S in Straight line ($180^\circ$); if two angles form a straight line, subtract from $180$ to find the other. Because both an angle and its supplement share the same sine value ($\sin(\pi-x)=\sin x$), solving $\sin x = k$ on $[0,\pi]$ always gives two solutions, one direct consequence of this supplementary relationship, which also underlies the supplementary opposite angles found in every cyclic quadrilateral.