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Angles

Identify and express angles using proper notation.

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 \({180}^{∘}.\) An angle measuring \(\frac{\pi }{2}\) radians, or 1.570796 radians, measures \({90}^{∘}.\) To translate degrees to radians, we multiply the angle measure in degrees by \(\frac{\pi }{180}.\) For example, to write \({45}^{∘}\) in radians, we have \[{45}^{∘}(\frac{\pi }{180})=\frac{\pi }{4}=0.785398\ \text{radians.}\]

To translate radians to degrees, we multiply by \(\frac{180}{\pi }.\) For example, to write \(2\pi\) radians in degrees, we have \[2\pi (\frac{180}{\pi })={360}^{∘}.\]

Another example of translating radians to degrees and degrees to radians is \(\frac{2\pi }{3}.\) To write in degrees, we have \(\frac{2\pi }{3}(\frac{180}{\pi })={120}^{∘}.\) To write \({30}^{∘}\) in radians, we have \({30}^{∘}(\frac{\pi }{180})=\frac{\pi }{6}\). 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 \({180}^{∘}\); a right angle measures \({90}^{∘};\) an acute angle is any angle whose measure is less than \({90}^{∘};\) and an obtuse angle is any angle whose measure is between \({90}^{∘}\) and \({180}^{∘}.\) See .

An easy way to measure angles is with a protractor (). 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 \(∠\) or \(∡\) before the points. When we are referring to the measure of the angle, we use the symbol \(m∡\). See .

We can name this angle \(∡BAC\), or \(∡CAB\), or \(∡A.\)

Classifying Angles

Try it.

Determine which angles are acute, right, obtuse, or straight on the graph (). You may want to use a protractor for this one.

Solution

Acute angles measure less than \({90}^{∘}.\)Obtuse angles measure between \({90}^{∘}\) and \({180}^{∘}.\)Right angles measure \({90}^{∘}.\)Straight angles measure \({180}^{∘}.\)
\(\begin{array}{l}∠EOF \\ ∠EOG \\ ∠FOG \\ ∠GOH \\ ∠FOH \\ ∠HOJ \\ ∠HOK \\ ∠JOK \\ ∠KOL \\ ∠JOL\end{array}\)\(\begin{array}{l}∠EOJ \\ ∠EOK \\ ∠FOJ \\ ∠FOK \\ ∠FOL \\ ∠GOK \\ ∠GOL\end{array}\)\(\begin{array}{l}∠EOH \\ ∠HOL \\ ∠GOJ\end{array}\)\(∠EOL\)

Most angles can be classified visually or by description. However, if you are unsure, use a protractor.

Adjacent Angles

Two angles with the same starting point or vertex and one common side are called adjacent angles. In , angle \(∠DBC\) is adjacent to \(∠CBA\). 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 \({180}^{∘}.\) In , we are given that \(m∡FBE={35}^{∘},\) so what is \(m∡ABE?\) These are supplementary angles. Therefore, because \(m∡ABF={180}^{∘}\), and as \({180}^{∘}-{35}^{∘}={145}^{∘},\) we have \(m∡ABE={145}^{∘}.\)

Solving for Angle Measurements and Supplementary Angles

Try it.

Solve for the angle measurements in .

Solution

Step 1: These are supplementary angles. We can see this because the two angles are part of a horizontal line, and a horizontal line represents \({180}^{∘}.\) Therefore, the sum of the two angles equals \({180}^{∘}.\)

Step 2: \[\begin{array}{lll}(32x-7)+(5x+2) & = & 180 \\ 37x-5 & = & 180 \\ 37x & = & 185 \\ x & = & 5\end{array}\]

Step 3: Find the measure of each angle: \[\begin{array}{lll}32x-7 & = & 32(5)-7 \\ & = & {153}^{∘} \\ 5x+2 & = & 5(5)+2 \\ & = & {27}^{∘}\end{array}\]

Step 4: We check: \({153}^{∘}+{27}^{∘}={180}^{∘}.\)

Complementary Angles

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

Solving for Angle Measurements and Complementary Angles

Try it.

Solve for the angle measurements in .

Solution

We have that \[\begin{array}{lll}(9x-5)+4x+(7x-5) & = & 90 \\ 20x & = & 100 \\ x & = & 5\end{array}\]

Then, \(m∡(9x-5)={40}^{∘}\), \(m∡(4x)={20}^{∘}\), and \(m∡(7x-5)={30}^{∘}.\)

Vertical Angles

When two lines intersect, the opposite angles are called vertical angles, and vertical angles have equal measure. For example, 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.

Calculating Vertical Angles

Try it.

In , one angle measures \({40}^{∘}.\) Find the measures of the remaining angles.

Solution

The 40-degree angle and \(∠2\) are vertical angles. Therefore, \(m∡2={40}^{∘}.\)

Notice that \(∡2\) and \(∡1\) are supplementary angles, meaning that the sum of \(m∡2\) and \(m∡1\) equals \({180}^{∘}.\) Therefore, \(m∡1={180}^{∘}-{40}^{∘}={140}^{∘}\).

Since \(∡1\) and \(∡3\) are vertical angles, then \(m∡3\) equals \({140}^{∘}.\)

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 . 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, \(∡3\) and \(∡6\) are alternate interior angles and have equal measure; \(∡4\) and \(∡5\) are alternate interior angles and have equal measure as well. See .

Alternate Exterior Angles

Alternate exterior angles are exterior angles on opposite sides of the transversal and have the same measure. For example, in , \(∡2\) and \(∡7\) are alternate exterior angles and have equal measures; \(∡1\) and \(∡8\) 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 , \(∡1\) and \(∡5\) are corresponding angles and have equal measures; \(∡3\) and \(∡7\) are corresponding angles and have equal measures; \(∡2\) and \(∡6\) are corresponding angles and have equal measures; \(∡4\) and \(∡8\) are corresponding angles and have equal measures as well.

Evaluating Space

Try it.

You live on the corner of First Avenue and Linton Street. You want to plant a garden in the far corner of your property () and fence off the area. However, the corner of your property does not form the traditional right angle. You learned from the city that the streets cross at an angle equal to \({150}^{∘}.\) What is the measure of the angle that will border your garden?

Solution

As the angle between Linton Street and First Avenue is \({150}^{∘},\) the supplementary angle is \({30}^{∘}.\) Therefore, the garden will form a \({30}^{∘}\) angle at the corner of your property.

Determining Angles Formed by a Transversal

Try it.

In given that angle 3 measures \({40}^{∘},\) find the measures of the remaining angles and give a reason for your solution.

Solution

\(m∡2=m∡3={40}^{∘}\) by vertical angles.

\(∡3=m∡7\) by corresponding angles.

\(m∡7=m∡6={40}^{∘}\) by vertical angles.

\(m∡1=180-40={140}^{∘}\) by supplementary angles.

\(m∡4=m∡1={140}^{∘}\) by vertical angles.

\(m∡8=m∡1={140}^{∘}\) by alternate exterior angles.

\(m∡5=m∡8={140}^{∘}\) by vertical angles.

Measuring Angles Formed by a Transversal

Try it.

In given that angle 2 measures \({23}^{∘},\) find the measure of the remaining angles and state the reason for your solution.

Solution

\(m∡2=m∡3={23}^{∘}\) by vertical angles, because \(∡2\) and \(∡3\) are the opposite angles formed by two intersecting lines.

\(m∡1={157}^{∘}\) by supplementary angles to \(m∡2\) or \(m∡3.\) We see that \(∡1\) and \(∡2\) form a straight angle as does \(∡1\) and \(∡3.\) A straight angle measures \({180}^{∘},\) so \({180}^{∘}-{23}^{∘}={157}^{∘}.\)

\(m∡4=m∡1={157}^{∘}\) by vertical angles, because \(∡4\) and \(∡1\) are the two opposite angles formed by two intersecting lines.

\(m∡5=m∡1={157}^{∘}\) by corresponding angles because they are the same angle formed by the transversal crossing two parallel lines, one exterior and one interior.

\(m∡8=m∡5={157}^{∘}\) by vertical angles because \(∡8\) and \(∡5\) are the two opposite angles formed by two intersecting lines.

\(m∡7=m∡2={23}^{∘}\) by alternate exterior angles because, like vertical angles, these angles are the opposite angles formed by the transversal intersecting two parallel lines.

\(m∡6=m∡7={23}^{∘}\) by vertical angles because these are the opposite angles formed by two intersecting lines.

Condensed — the full section is in OpenStax Contemporary Mathematics.

Key Concepts

  • Angles are classified as acute if they measure less than \({90}^{∘},\) obtuse if they measure greater than \({90}^{∘}\) and less than \({180}^{∘},\) right if they measure exactly \({90}^{∘},\) and straight if they measure exactly \({180}^{∘}.\)
  • If the sum of angles equals \({90}^{∘}\), they are complimentary angles. If the sum of angles equals \({180}^{∘}\), 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 \(\frac{180}{\pi }.\)

Practice (8)

Try each one on paper first. Reveal the answer to check; verified ones can be opened in the solver for every step.

  1. Determine which angles are acute, right, obtuse, or straight on the graph (). You may want to use a protractor for this one.

    ଉତ୍ତରକୁ ଖୋଲନ୍ତୁ

    Acute angles measure less than \({90}^{∘}.\)Obtuse angles measure between \({90}^{∘}\) and \({180}^{∘}.\)Right angles measure \({90}^{∘}.\)Straight angles measure \({180}^{∘}.\)
    \(\begin{array}{l}∠EOF \\ ∠EOG \\ ∠FOG \\ ∠GOH \\ ∠FOH \\ ∠HOJ \\ ∠HOK \\ ∠JOK \\ ∠KOL \\ ∠JOL\end{array}\)\(\begin{array}{l}∠EOJ \\ ∠EOK \\ ∠FOJ \\ ∠FOK \\ ∠FOL \\ ∠GOK \\ ∠GOL\end{array}\)\(\begin{array}{l}∠EOH \\ ∠HOL \\ ∠GOJ\end{array}\)\(∠EOL\)

    Most angles can be classified visually or by description. However, if you are unsure, use a protractor.

  2. Solve for the angle measurements in .

    ଉତ୍ତରକୁ ଖୋଲନ୍ତୁ

    Step 1: These are supplementary angles. We can see this because the two angles are part of a horizontal line, and a horizontal line represents \({180}^{∘}.\) Therefore, the sum of the two angles equals \({180}^{∘}.\)

    Step 2: \[\begin{array}{lll}(32x-7)+(5x+2) & = & 180 \\ 37x-5 & = & 180 \\ 37x & = & 185 \\ x & = & 5\end{array}\]

    Step 3: Find the measure of each angle: \[\begin{array}{lll}32x-7 & = & 32(5)-7 \\ & = & {153}^{∘} \\ 5x+2 & = & 5(5)+2 \\ & = & {27}^{∘}\end{array}\]

    Step 4: We check: \({153}^{∘}+{27}^{∘}={180}^{∘}.\)

  3. Solve for the angle measurements in .

    ଉତ୍ତରକୁ ଖୋଲନ୍ତୁ

    We have that \[\begin{array}{lll}(9x-5)+4x+(7x-5) & = & 90 \\ 20x & = & 100 \\ x & = & 5\end{array}\]

    Then, \(m∡(9x-5)={40}^{∘}\), \(m∡(4x)={20}^{∘}\), and \(m∡(7x-5)={30}^{∘}.\)

  4. In , one angle measures \({40}^{∘}.\) Find the measures of the remaining angles.

    ଉତ୍ତରକୁ ଖୋଲନ୍ତୁ

    The 40-degree angle and \(∠2\) are vertical angles. Therefore, \(m∡2={40}^{∘}.\)

    Notice that \(∡2\) and \(∡1\) are supplementary angles, meaning that the sum of \(m∡2\) and \(m∡1\) equals \({180}^{∘}.\) Therefore, \(m∡1={180}^{∘}-{40}^{∘}={140}^{∘}\).

    Since \(∡1\) and \(∡3\) are vertical angles, then \(m∡3\) equals \({140}^{∘}.\)

  5. You live on the corner of First Avenue and Linton Street. You want to plant a garden in the far corner of your property () and fence off the area. However, the corner of your property does not form the traditional right angle. You learned from the city that the streets cross at an angle equal to \({150}^{∘}.\) What is the measure of the angle that will border your garden?

    ଉତ୍ତରକୁ ଖୋଲନ୍ତୁ

    As the angle between Linton Street and First Avenue is \({150}^{∘},\) the supplementary angle is \({30}^{∘}.\) Therefore, the garden will form a \({30}^{∘}\) angle at the corner of your property.

  6. In given that angle 3 measures \({40}^{∘},\) find the measures of the remaining angles and give a reason for your solution.

    ଉତ୍ତରକୁ ଖୋଲନ୍ତୁ

    \(m∡2=m∡3={40}^{∘}\) by vertical angles.

    \(∡3=m∡7\) by corresponding angles.

    \(m∡7=m∡6={40}^{∘}\) by vertical angles.

    \(m∡1=180-40={140}^{∘}\) by supplementary angles.

    \(m∡4=m∡1={140}^{∘}\) by vertical angles.

    \(m∡8=m∡1={140}^{∘}\) by alternate exterior angles.

    \(m∡5=m∡8={140}^{∘}\) by vertical angles.

  7. In given that angle 2 measures \({23}^{∘},\) find the measure of the remaining angles and state the reason for your solution.

    ଉତ୍ତରକୁ ଖୋଲନ୍ତୁ

    \(m∡2=m∡3={23}^{∘}\) by vertical angles, because \(∡2\) and \(∡3\) are the opposite angles formed by two intersecting lines.

    \(m∡1={157}^{∘}\) by supplementary angles to \(m∡2\) or \(m∡3.\) We see that \(∡1\) and \(∡2\) form a straight angle as does \(∡1\) and \(∡3.\) A straight angle measures \({180}^{∘},\) so \({180}^{∘}-{23}^{∘}={157}^{∘}.\)

    \(m∡4=m∡1={157}^{∘}\) by vertical angles, because \(∡4\) and \(∡1\) are the two opposite angles formed by two intersecting lines.

    \(m∡5=m∡1={157}^{∘}\) by corresponding angles because they are the same angle formed by the transversal crossing two parallel lines, one exterior and one interior.

    \(m∡8=m∡5={157}^{∘}\) by vertical angles because \(∡8\) and \(∡5\) are the two opposite angles formed by two intersecting lines.

    \(m∡7=m∡2={23}^{∘}\) by alternate exterior angles because, like vertical angles, these angles are the opposite angles formed by the transversal intersecting two parallel lines.

    \(m∡6=m∡7={23}^{∘}\) by vertical angles because these are the opposite angles formed by two intersecting lines.

  8. Find the measures of the angles 1, 2, 4, 11, 12, and 14 in and the reason for your answer given that \({l}_{1}\) and \({l}_{2}\) are parallel.

    ଉତ୍ତରକୁ ଖୋଲନ୍ତୁ

    \(m∡12={118}^{∘}\), supplementary angles

    \(m∡14={118}^{∘}\), vertical angles

    \(m∡11={62}^{∘}\), vertical angles

    \(m∡4={62}^{∘}\), corresponding angles

    \(m∡1={62}^{∘}\), vertical angles

    \(m∡2={56}^{∘}\), supplementary angles

Symbols used here

\pi
pi
Ratio of a circle's circumference to its diameter, 3.14159…
^\circ
degrees
1/360 of a full turn. 180° = π radians.
\theta
theta
The usual name for an angle.
\sin,\ \cos,\ \tan
sine, cosine, tangent
Ratios of sides in a right triangle; coordinates on the unit circle.
\angle ABC,\ \triangle ABC
angle, triangle
The angle at B between BA and BC; the triangle with those vertices.
\parallel,\ \perp,\ \cong,\ \sim
parallel, perpendicular, congruent, similar
Never meet; meet at 90°; identical shape and size; same shape.

How to: Angles

  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.
  6. vertex
  7. right angle
  8. acute angle

Questions people ask

Why does every triangle have angles adding to 180°?

Draw a line through one vertex parallel to the opposite side: the two alternate angles equal the other two angles of the triangle, and the three angles at the vertex lie on a straight line. That is Euclid's proof, and it is why the fact holds only on a flat plane.

When do I use the law of sines versus the law of cosines?

Cosines when you know three sides, or two sides and the angle between them. Sines when you know an angle and the side opposite it, plus one more piece.

What is the difference between area and perimeter?

Perimeter is the length of the boundary (one dimension, measured in metres); area is the amount of surface inside (two dimensions, square metres). Doubling every side doubles the perimeter but quadruples the area.

ନିଜେ ଚେଷ୍ଟାକରନ୍ତୁ

Parts of this page are adapted from OpenStax Contemporary Mathematics (CC BY-NC-SA 4.0). Condensed and re-explained here; errors are ours.

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