Introduction
Rays are are just part of a line with an endpoint. Rays are named with two points, the endpoint being first. For example, the following is Ray $ AB \,( \overrightarrow{{AB}})$, or Ray $ AC \,( \overrightarrow{{AC}})$:
Angles are formed with two non-collinear rays, which are called the sides of the angles. The endpoint ($ C$) is the vertex of the angle. Thus, point $ C$ is the vertex for angles $ \angle ACD$, $ \angle ACB$, or $ \angle DCB$ (or by reversing the first and last letters). We can also call an angle by it’s vertex only, but only if there is exactly one angle with that vertex.
Note that points on a plane can be located in the interior of an angle, on the angle, or on the exterior of an angle.
Measuring Angles
Angles are measured in units that are called degrees, which are derived by dividing a circle into $ 360 $ parts. Have you ever heard the expression “made a $ 180$? It means reversing one’s course, which, as we’ll see later, is in the opposite direction of $ 0$ degrees. A degree is symbolized with a small circle, such as $ 180{}^\circ $ (yes, just like temperature degrees!)
A protractor can be used to measure degrees. To measure, I find it easiest to begin with the right-hand side at $ 0{}^\circ $ going counter-clockwise to see where the angle ends up. For example, $ \angle DCE=40{}^\circ $ (acute angle, since it measures $ < 90{}^\circ $), $ \angle BCE=90{}^\circ $ (right angle, since it measures $ 90{}^\circ $), and $ \angle ACE=120{}^\circ $ (obtuse angle, since it measures $ > 90{}^\circ $). Notice the square symbol near on the vertex to indicate the right angle.
Angle Relationships
Here are some angle relationship definitions you should be familiar with:
- Angle Bisector. A ray that divides an angle into two congruent (same measurement) angles.
- Adjacent Angles: Angles that share a ray that divides them. They have a common vertex and are one plane, but have no common interior points.
- Linear Pair: A pair of adjacent angles that are “on a line” (noncommon sides are opposite rays).
- Vertical Angles: Two (non-adjacent) angles formed by two intersection lines.
- Complementary Angles:
- Supplementary Angles:
- Perpendicular Lines:
Problems and Solutions