An angle is the union of two rays with a common endpoint, called the vertex. The angles formed by vertical and horizontal lines are called right angles; lines, segments, or rays that intersect in right angles are said to be perpendicular.
Angles, for our purposes, can be measured in either degrees (from 0 to 360) or radians (from 0 to ). Angles length can be determined by measuring along the arc they map out on a circle. In radians we consider the length of the arc of the circle mapped out by the angle. Since the circumference of a circle is , a right angle is radians. In degrees, the circle is 360 degrees, and so a right angle would be 90 degrees.
Angles are named in several ways.
- By naming the vertex of the angle (only if there is only one angle formed at that vertex; the name must be non-ambiguous)
- By naming a point on each side of the angle with the vertex in between.
- By placing a small number on the interior of the angle near the vertex.
Classification of Angles by Degree Measure
- an angle is said to be acute if it measures between 0 and 90 degrees, exclusive.
- an angle is said to be right if it measures 90 degrees.
- notice the small box placed in the corner of a right angle, unless the box is present it is not assumed the angle is 90 degrees.
- all right angles are congruent
- an angle is said to be obtuse if it measures between 90 and 180 degrees, exclusive.
Special Pairs of Angles
- adjacent angles
- adjacent angles are angles with a common vertex and a common side.
- adjacent angles have no interior points in common.
- complementary angles
- complementary angles are two angles whose sum is 90 degrees.
- complementary angles may or may not be adjacent.
- if two complementary angles are adjacent, then their exterior sides are perpendicular.
- supplementary angles
- two angles are said to be supplementary if their sum is 180 degrees.
- supplementary angles need not be adjacent.
- if supplementary angles are adjacent, then the sides they do not share form a line.
- linear pair
- if a pair of angles is both adjacent and supplementary, they are said to form a linear pair.
- vertical angles
- angles with a common vertex whose sides form opposite rays are called vertical angles.
- vertical angles are congruent.