Measurements of Common Angles

Full Angle
A full angle is equal to one full rotation.

From Full Rotation is 2 pi radians, a full rotation is $$2\pi$$ radians. From the definition of degrees, a full rotation is $$360^\circ$$.

Therefore, a full angle is $$360^\circ$$ or $$2\pi$$.

Straight Angle
A full rotation is defined to be $$360^\circ$$ and is provably $$2\pi$$ radians.

Since lines are straight, it therefore follows that from any point on a line, the angle between one side of the line and the other is one half of a full rotation. Therefore, the measurement of a straight angle is $$\frac{360^\circ}{2}=180^\circ$$ or $$\frac{2\pi}{2}=\pi$$.

Right Angle
Since right angles are equal to one half of a straight angle, the measurement of a right angle is $$\frac{180^\circ}{2}=90^\circ$$ or $$\frac{\pi}{2}$$.

Acute Angle
An acute angle is defined to be an angle whose measure is between that of a zero angle and a right angle. Since a zero angle measures $$0$$ and a right angle measures $$90^\circ$$ or $$\frac{\pi}{2}$$, it follows that an acute angle measures $$\theta$$, where $$0^\circ<\theta<90^\circ$$ or $$0<\theta<\frac{\pi}{2}$$.

Obtuse Angle
An obtuse angle is defined to be an angle whose measure is between that of a right angle and a straight angle. Since a right angle measures $$90^\circ$$ or $$\frac{\pi}{2}$$ and a straight angle measures $$180^\circ$$ or $$\pi$$, it follows that an acute angle measures $$\theta$$, where $$90^\circ<\theta<180^\circ$$ or $$\frac{\pi}{2}<\theta<\pi$$.

Reflex Angle
A reflex angle is defined to be an angle whose measure is between that of a straight angle and a full angle. Since a straight angle measures $$180^\circ$$ or $$\pi$$ and a full angle measures $$360^\circ$$ or $$2\pi$$, it follows that a reflex angle measures $$\theta$$, where $$180^\circ<\theta<360^\circ$$ or $$\pi<\theta<2\pi$$.