Maths 2: Angles and Angle Measurement

Definition of an Angle

A figure is an angle if and only if it is the union of two non-collinear rays, the sides, with a common endpoint, the vertex.

Definition of Adjacent Angles

Two angles are said to be adjacent if and only if they satisfy the following three conditions:
1. they have a common vertex
2. they have a common side
3. they DO NOT have common interior points

Angle Measure Postulate (AMP)

Given an angle, there is a unique real number BETWEEN 0 and 180 known as its degree measure. (important: this explains why there is no 0 and 180 angles)

Protractor Postulate

In a half-plane with edge, line AB, and any point S between A and B, there is a one-to-one correspondence between the rays that originate at S in that half-plane and the real numbers BETWEEN 0 and 180. The angle measure is taken as the absolute value between the two corresponding real numbers.

Theorem

In a half-plane, through the endpoint of a ray lying in the edge of the half-plane, there is EXACTLY one other ray such that the angle formed by the two rays has a given measure between 0 and 180.

Definition of Angle Congruence

Two angles are said to be congruent if and only if their measures are equal.

Theorem

All right angles are congruent.

Definition of an Angle Bisector

Given three coplanar rays, OA, OT and OB, ray OT is the angle bisector if and only if angle AOT is congruent to angle TOB