Geometry/Angles

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 $2\pi$). 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 $2\pi$, a right angle is ${\frac {\pi }{2}}$ radians. In degrees, the circle is 360 degrees, and so a right angle would be 90 degrees.

Naming Conventions

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) $\angle B$
• By naming a point on each side of the angle with the vertex in between. $\angle ABC$
• By placing a small number on the interior of the angle near the vertex.

Classification of Angles by Degree Measure

Acute Angle

• an angle is said to be acute if it measures between 0 and 90 degrees, exclusive.

Right Angle

• 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

Obtuse Angle

• 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.