Straight Angle
A straight angle measures exactly 180 degrees and looks like a straight line.
Formula
\text{angle} = 180^\circ = \pi \text{ radians}
Definition
A straight angle is exactly $180^\circ$ ($\pi$ radians): its two sides point in exactly opposite directions, forming a straight line through the vertex, and it is the boundary between obtuse angles (less than $180^\circ$) and reflex angles (greater than $180^\circ$). If ray $AB$ and ray $AC$ form a straight angle, then $B$, $A$, and $C$ are collinear, and in the complex plane, multiplying by $e^{i\pi} = -1$ represents a rotation by a straight angle, i.e. a reflection through the origin.
Example
Stretching both arms straight out to the sides creates a straight angle from one hand to the other through your body, and the top edge of a desk shows one too. A linear pair of angles always sums to a straight angle ($180^\circ$), so the supplement of any angle is whatever is needed to reach $180^\circ$. Euler's identity $e^{i\pi} + 1 = 0$ encodes the straight angle as a rotation: $e^{i\pi}$ rotates any complex number by $180^\circ$, negating it.
Key Insight
A straight angle is half of a full $360^\circ$ circle, and when two angles together form one, they are called supplementary angles, the basis for the exterior angle theorem and many other Euclidean proofs. The straight angle ($\pi$ radians) appears throughout mathematics: it is the period of the tangent function, the angle subtended by a diameter in a semicircle, and the rotation in Euler's formula, a ubiquity that reflects the fundamental role of $\pi$ in circular geometry.