Angle

An angle is formed by two rays sharing a common endpoint called the vertex. The size of the angle measures how much one ray would rotate to coincide with the other.

Two units of measurement are standard:

• Degrees: a full circle is $360°$. Right angle = $90°$, straight angle = $180°$.
• Radians: a full circle is $2\pi$. Conversion: $1\text{ rad} = \frac{180}{\pi}° \approx 57.296°$.

Radians are the "natural" unit for calculus because $\frac{d}{dx}\sin x = \cos x$ holds only when $x$ is in radians.

Angle classifications:

• Acute: less than $90°$,
• Right: exactly $90°$,
• Obtuse: between $90°$ and $180°$,
• Straight: $180°$,
• Reflex: greater than $180°$.

Pairs of angles can be complementary (sum $= 90°$), supplementary ($= 180°$), or vertical (across a crossing of two lines, always equal).