Straight Angle
GeometryA 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 $180^\circ$. Its two sides point in exactly opposite directions, forming a straight line. A straight angle looks just like a flat, straight line.
Example
When you stretch both arms straight out to the sides, the angle from one hand to the other through your body is a straight angle. The top edge of your desk forms a straight angle.
Key Insight
A straight angle is half of a full $360^\circ$ circle. When two angles together form a straight angle (a line), they are called supplementary angles - they add up to $180^\circ$.
Definition
A straight angle measures exactly $180^\circ$ ($\pi$ radians). Its two rays point in exactly opposite directions, forming a straight line through the vertex. A straight angle is the boundary between obtuse angles (less than $180^\circ$) and reflex angles (greater than $180^\circ$).
Example
If ray $AB$ and ray $AC$ form a straight angle, then $B$, $A$, and $C$ are collinear (they lie on the same line). A linear pair of angles always sums to a straight angle ($180^\circ$). The supplement of any angle is the amount needed to reach $180^\circ$.
Key Insight
The straight angle is why supplementary angles sum to $180^\circ$: two adjacent supplementary angles together form a straight angle (a flat line). This is the basis for the exterior angle theorem and many other proofs in Euclidean geometry.
Definition
A straight angle is $\pi$ radians. In the complex plane, multiplying by $e^{i\pi} = -1$ (Euler's identity) represents a rotation by a straight angle, i.e., reflection through the origin. Two opposite rays from a point partition the plane into two half-planes.
Example
Euler's identity $e^{i\pi} + 1 = 0$ encodes the straight angle as a rotation: $e^{i\pi}$ rotates any complex number by $180^\circ$ ($\pi$ radians), negating it. This connects the straight angle to the deepest identity in complex analysis.
Key Insight
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. Its ubiquity reflects the fundamental role of $\pi$ in circular geometry.