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!).
Note: to indicate an angle’s measurement, we typically use a small “$ m$” before, such as $ m\angle DCE=40{}^\circ $; this indicates a number. When we say two angles are congruent, we are talking about a size, so we just use the name of the angle, such as $ \angle ABC \displaystyle \cong \angle DEF$.
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, $ m\angle DCE=40{}^\circ $ (acute angle, since it measures $ < 90{}^\circ $), $ m\angle BCE=90{}^\circ $ (right angle, since it measures $ 90{}^\circ $), and $ m\angle ACE=120{}^\circ $ (obtuse angle, since it measures $ > 90{}^\circ $). Notice the square symbol near on the vertex to indicate the right angle. Also notice that angle measure of $ 180{}^\circ $ is known as a straight angle, and forms a line.
Angle Relationships
Here are some angle relationship definitions you should be familiar with:
- Angle Bisector. A ray, segment, or line that divides an angle into two congruent (same measurement) angles. $\overrightarrow{{CD}}$ bisects $ \angle BCE$; thus, $ \angle BCD \displaystyle \cong \angle DCE$. Note the notation to show congruent angles ($ \displaystyle \cong$) and in the following figure:
- 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. $ \angle BCD$ and $ \angle DCE$ above are adjacent.
- Linear Pair: A pair of adjacent angles that are “on a line” (noncommon sides are opposite rays). In the following figure, angles $ \angle BCD$ and $ \angle DCE$ form a linear pair; also note that $ m\angle BCD+m\angle DCE=180{}^\circ $.
- Vertical Angles: Two (non-adjacent) angles formed by two intersection lines. In the following figure, angles $ \angle 1$ and $ \angle 3$ are vertical, as well as angles $ \angle 2$ and $ \angle 4$. Also note that, as it looks, $ \angle 1 \displaystyle \cong \angle 3$ and $ \angle 2 \displaystyle \cong \angle 4$, since they are directly opposite each other. Remember, though, in Geometry, you can’t always rely on how things look.
- Complementary Angles: Two angles whose measurements add up to $ 90{}^\circ $. The angles don’t have to be adjacent angles, but they can. In this figure, adjacent angles $ \angle BCD$ and $ \angle DCE$ are complementary since they add up to $ 90{}^\circ $ (are also adjacent, so they form a right angle).
- Supplementary Angles: Two angles whose measurements add up to $ 180{}^\circ $. The angles don’t have to be adjacent angles, but they can, such as a linear pair. In this figure, adjacent angles $ \angle 1$ and $ \angle 2$ are supplementary since they add up to $ 180{}^\circ $. It so happens that the angles form a straight line, and thus are also a linear pair.
- Perpendicular Lines: Lines, line segments, or rays forming right angles are perpendicular (denoted by the $ \displaystyle \bot$ symbol). In the following figure, $ \displaystyle \overleftrightarrow{{BD}}\bot\, \overleftrightarrow{{EC}}$:
Problems and Solutions
Problem: Find $ x$ and $ y$ in the following figure:
Solution: First solve for $ y$, since $ y+5y=180$ (linear pair): $ 6y=180;\,y=30$. Now we know that $ 2x-10=30$, since these are vertical angles. Thus, $ 2x=40;\,x=20$. Try it; it works! They always love to throw in Algebra in Geometry, so you won’t forget it!
Problem: The ratio of measures of two complementary angles is $ \displaystyle 3:2$. What are the measures of the angles?
Solution: Complementary angles add up to $ 90{}^\circ $, so let $ x=$ the first angle and $ 90-x=$ the second angle. Since the ratio of the measure of first angle to the second angle is $ \displaystyle 3:2$, set up the ratios as proportions (fractions), and cross-multiply: $ \displaystyle \frac{3}{2}=\frac{x}{{90-x}}$. We get:
$ \displaystyle \begin{array}{c}\,\displaystyle \frac{3}{2}=\displaystyle \frac{x}{{90-x}}\,\\\,3\left( {90-x} \right)=2x\\270-3x=2x\\270=5x\\x=54{}^\circ ;\,\,\,\,\,90-x=90-54=36{}^\circ \end{array}$
The two angles are $ 54{}^\circ $ and $ 36{}^\circ $. This makes sense since they add up to $ 90{}^\circ $ and the two measures are in a ratio of $ \displaystyle \frac{3}{2}$!
Problem: Answer the questions below for the following figure:
Name an angle or angle pair that meets the conditions. Note that there may be more than one correct answer.
- An angle complementary to $ \angle HFI$. Solution: $ \angle EFH$, since the two angles form a right angle, and thus add up to $ 90{}^\circ $. These are also adjacent angles.
- An angle supplementary to $ \angle ABE$. Solution: $ \angle CBE$, since the two angles form a line, and thus add up to $ 180{}^\circ $. This is also a linear pair, and the two angles are adjacent.
- Two complementary but not adjacent angles. Solution: $ \angle BAE$ and $ \angle BCF$, since the two angle measurements add up to $ 90{}^\circ $.
- Two acute vertical angles. Solution: $ \angle BEA$ and $ \angle FEH$, since the angles are across from each other and each measures $ <90{}^\circ $.
- Two obtuse vertical angles. Solution: $ \angle EHF$ and $ \angle GHI$, since the angles are across from each other and each measures $ >90{}^\circ $.
- An angle bisector. Solution: $ \overline{{AF}}$, since it divides $ \angle CFI$ (straight angle, or line) into two congruent (right) angles.