Angle

Dependency: Protractor axiom. A derived definition: how widely two sides open is reduced to a real number supplied by the protractor axiom.

Definition

Draw two rays $OA$ and $OB$ from the same point $O$.

The figure formed by these two rays is called an angle and is denoted by $\angle AOB$. $O$ is its vertex, and rays $OA$ and $OB$ are its sides.

The size of an angle is not estimated by sight. The Protractor axiom axiom assigns every angle a real number from $0^\circ$ to $180^\circ$, called its degree measure and also denoted by $\angle AOB$. Context tells whether the notation means the geometric figure or its measure. Thus equality of angles is equality of real numbers, which can be added, compared, and calculated.

Remarks

• Only the opening matters. The measure depends only on how far the sides open. Extending or shortening a drawing, or scaling the entire figure, does not change it. That is why the sides are rays rather than segments.
• Special measures. $90^\circ$ is a right angle, with perpendicular sides. An angle below $90^\circ$ is acute; one between $90^\circ$ and $180^\circ$ is obtuse; and $180^\circ$ is a straight angle.
• One notation, two roles. In $\angle AOB=\angle A'O'B'$, the notation refers to measures; in “inside $\angle AOB$,” it refers to the figure.
• Angle is one of geometry's basic measurements. To decide whether figures are congruent or similar, we compare corresponding side lengths and angle measures.