Supplementary Angles
GeometrySupplementary 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 $180^\circ$ - a straight line. You can think of them as two pieces that together form a straight angle.
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.
Key Insight
A memory trick: S in Supplementary goes with S in Straight line ($180^\circ$). If you know one angle and the two angles form a straight line, subtract from $180$ to find the other.
Definition
Two angles are supplementary if and only if their measures sum to $180^\circ$. Adjacent supplementary angles form a linear pair. Co-interior angles (same-side interior angles) formed by a transversal crossing parallel lines are always supplementary.
Example
Angles of $125^\circ$ and $55^\circ$ are supplementary. When parallel lines are cut by a transversal, co-interior angles are supplementary: if one is $115^\circ$, the other is $65^\circ$. The supplement of an obtuse angle is always acute.
Key Insight
Supplementary angles are the basis for the co-interior angles theorem: when a transversal cuts parallel lines, same-side interior angles are supplementary. This theorem is used to prove lines are parallel and to find unknown angles.
Definition
Angles $\alpha$ and $\beta$ are supplementary iff $\alpha + \beta = \pi$. The identity $\sin(\pi - x) = \sin x$ and $\cos(\pi - x) = -\cos x$ express the analytic version of supplementarity. In a cyclic quadrilateral (inscribed in a circle), opposite angles are supplementary - a consequence of the inscribed angle theorem.
Example
In cyclic quadrilateral $ABCD$: angle $A$ + angle $C$ $= 180^\circ$, angle $B$ + angle $D$ $= 180^\circ$. This property characterizes cyclic quadrilaterals: a quadrilateral can be inscribed in a circle iff its opposite angles are supplementary.
Key Insight
The supplementary angle relationship in cyclic quadrilaterals (Ptolemy's theorem context) and the identity $\sin(\pi - x) = \sin x$ together explain why two different angles can have the same sine value: any angle and its supplement share a sine value, giving two solutions in $[0, \pi]$ when solving $\sin x = k$.