10.2 Angles

Learning Objectives

After completing this section, you should be able to:

1. Identify and express angles using proper notation.
2. Classify angles by their measurement.
3. Solve application problems involving angles.
4. Compute angles formed by transversals to parallel lines.
5. Solve application problems involving angles formed by parallel lines.

Classifying Angles

Angles are measured in radians or degrees. For example, an angle that measures $\pi$ radians, or 3.14159 radians, is equal to the angle measuring ${180}^{\circ }.$ An angle measuring $\frac{\pi }{2}$ radians, or 1.570796 radians, measures ${90}^{\circ }.$ To translate degrees to radians, we multiply the angle measure in degrees by $\frac{\pi }{180}.$ For example, to write ${45}^{\circ }$ in radians, we have

$${45}^{\circ }\left(\frac{\pi }{180}\right)=\frac{\pi }{4}=0.785398\text{radians.}$$

To translate radians to degrees, we multiply by $\frac{180}{\pi }.$ For example, to write $2\pi$ radians in degrees, we have

$$2\pi \left(\frac{180}{\pi }\right)={360}^{\circ }.$$

Another example of translating radians to degrees and degrees to radians is $\frac{2\pi }{3}.$ To write in degrees, we have $\frac{2\pi }{3}\left(\frac{180}{\pi }\right)={120}^{\circ }.$ To write ${30}^{\circ }$ in radians, we have ${30}^{\circ }\left(\frac{\pi }{180}\right)=\frac{\pi }{6}$. However, we will use degrees throughout this chapter.

Several angles are referred to so often that they have been given special names. A straight angle measures ${180}^{\circ }$; a right angle measures ${90}^{\circ };$ an acute angle is any angle whose measure is less than ${90}^{\circ };$ and an obtuse angle is any angle whose measure is between ${90}^{\circ }$ and ${180}^{\circ }.$ See Figure 10.23.

An easy way to measure angles is with a protractor (Figure 10.24). A protractor is a very handy little tool, usually made of transparent plastic, like the one shown here.

With a protractor, you line up the straight bottom with the horizontal straight line of the angle. Be sure to have the center hole lined up with the vertex of the angle. Then, look for the mark on the protractor where the second ray lines up. As you can see from the image, the degrees are marked off. Where the second ray lines up is the measurement of the angle.

Notation

Naming angles can be done in couple of ways. We can name the angle by three points, one point on each of the sides and the vertex point in the middle, or we can name it by the vertex point alone. Also, we can use the symbols $\angle$ or $\measuredangle$ before the points. When we are referring to the measure of the angle, we use the symbol $m\measuredangle$. See Figure 10.25.

We can name this angle $\measuredangle BAC$, or $\measuredangle CAB$, or $\measuredangle A.$

Adjacent Angles

Two angles with the same starting point or vertex and one common side are called adjacent angles. In Figure 10.27, angle $\angle DBC$ is adjacent to $\angle CBA$. Notice that the way we designate an angle is with a point on each of its two sides and the vertex in the middle.

Supplementary Angles

Two angles are supplementary if the sum of their measures equals ${180}^{\circ }.$ In Figure 10.28, we are given that $m\measuredangle FBE={35}^{\circ },$ so what is $m\measuredangle ABE?$ These are supplementary angles. Therefore, because $m\measuredangle ABF={180}^{\circ }$, and as ${180}^{\circ }-{35}^{\circ }={145}^{\circ },$ we have $m\measuredangle ABE={145}^{\circ }.$

Complementary Angles

Two angles are complementary if the sum of their measures equals ${90}^{\circ }.$ In Figure 10.30, we have $m\measuredangle ABC={30}^{\circ },$ and $m\measuredangle ABD={90}^{\circ }.$ What is the $m\measuredangle CBD?$ These are complementary angles. Therefore, because ${90}^{\circ }-{30}^{\circ }={60}^{\circ },$ the $\measuredangle CBD={60}^{°}.$

Vertical Angles

When two lines intersect, the opposite angles are called vertical angles, and vertical angles have equal measure. For example, Figure 10.32 shows two straight lines intersecting each other. One set of opposite angles shows angle markers; those angles have the same measure. The other two opposite angles have the same measure as well.

Transversals

When two parallel lines are crossed by a straight line or transversal, eight angles are formed, including alternate interior angles, alternate exterior angles, corresponding angles, vertical angles, and supplementary angles. See Figure 10.34. Angles 1, 2, 7, and 8 are called exterior angles, and angles 3, 4, 5, and 6 are called interior angles.

Alternate Interior Angles

Alternate interior angles are the interior angles on opposite sides of the transversal. These two angles have the same measure. For example, $\measuredangle 3$ and $\measuredangle 6$ are alternate interior angles and have equal measure; $\measuredangle 4$ and $\measuredangle 5$ are alternate interior angles and have equal measure as well. See Figure 10.35.

Alternate Exterior Angles

Alternate exterior angles are exterior angles on opposite sides of the transversal and have the same measure. For example, in Figure 10.36, $\measuredangle 2$ and $\measuredangle 7$ are alternate exterior angles and have equal measures; $\measuredangle 1$ and $\measuredangle 8$ are alternate exterior angles and have equal measures as well.

Corresponding Angles

Corresponding angles refer to one exterior angle and one interior angle on the same side as the transversal, which have equal measures. In Figure 10.37, $\measuredangle 1$ and $\measuredangle 5$ are corresponding angles and have equal measures; $\measuredangle 3$ and $\measuredangle 7$ are corresponding angles and have equal measures; $\measuredangle 2$ and $\measuredangle 6$ are corresponding angles and have equal measures; $\measuredangle 4$ and $\measuredangle 8$ are corresponding angles and have equal measures as well.

Key Terms

• vertex
• right angle
• acute angle
• obtuse angle
• straight angle
• complementary
• supplementary

Key Concepts

• Angles are classified as acute if they measure less than ${90}^{\circ },$ obtuse if they measure greater than ${90}^{\circ }$ and less than ${180}^{\circ },$ right if they measure exactly ${90}^{\circ },$ and straight if they measure exactly ${180}^{\circ }.$
• If the sum of angles equals ${90}^{\circ }$, they are complimentary angles. If the sum of angles equals ${180}^{\circ }$, they are supplementary.
• A transversal crossing two parallel lines form a series of equal angles: alternate interior angles, alternate exterior angles, vertical angles, and corresponding angles

Formula

To translate an angle measured in degrees to radians, multiply by $\frac{\pi }{180}\text{.}$

To translate an angle measured in radians to degrees, multiply by $\frac{180}{\pi }.$