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 $\dfrac{360^\circ}{2}=180^\circ$ or $\dfrac{2\pi}{2}=\pi$.

See Two Angles making Two Right Angles make a Straight Line.

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

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 $\dfrac{\pi}{2}$, it follows that an acute angle measures $\theta$, where $0^\circ<\theta<90^\circ$ or $0<\theta<\dfrac{\pi}{2}$.

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 $\dfrac{\pi}{2}$ and a straight angle measures $180^\circ$ or $\pi$, it follows that an acute angle measures $\theta$, where $90^\circ<\theta<180^\circ$ or $\dfrac{\pi}{2}<\theta<\pi$.

Reflex Angle
A reflex angle is defined to be an angle whose measure is between that of a straight angle and a full angle.

From Two Angles making Two Right Angles make a Straight Line, a straight angle measures $180^\circ$ or $\pi$.

We also have that a full angle measures $360^\circ$ or $2\pi$.

It follows that a reflex angle measures $\theta$, where $180^\circ < \theta < 360^\circ$ or $\pi < \theta < 2 \pi$.