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10.3: Angles

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    129641
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    The exterior view of an architectural building.
    Figure 10.26 This modern architectural design emphasizes sharp reflective angles as part of the aesthetic through the use of glass walls. (credit: “Société Générale @ La Défense @ Paris” by Images Guilhem Vellut/Flickr, CC BY 2.0)

    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.

    Unusual perspectives on architecture can reveal some extremely creative images. For example, aerial views of cities reveal some exciting and unexpected angles. Add reflections on glass or steel, lighting, and impressive textures, and the structure is a work of art. Understanding angles is critical to many fields, including engineering, architecture, landscaping, space planning, and so on. This is the topic of this section.

    We begin our study of angles with a description of how angles are formed and how they are classified. An angle is the joining of two rays, which sweep out as the sides of the angle, with a common endpoint. The common endpoint is called the vertex. We will often need to refer to more than one vertex, so you will want to know the plural of vertex, which is vertices.

    In Figure 10.27, let the ray ABAB stay put. Rotate the second ray ACAC in a counterclockwise direction to the size of the angle you want. The angle is formed by the amount of rotation of the second ray. When the ray ACAC continues to rotate in a counterclockwise direction back to its original position coinciding with ray AB,AB, the ray will have swept out 360.360. We call the rays the “sides” of the angle.

    Two rays, A B and A C make an acute angle. A point, C is marked on the ray, A B. An arrow from A B points to A C.
    Figure 10.27 Vertex and Sides of an Angle

    Classifying Angles

    Angles are measured in radians or degrees. For example, an angle that measures ππ radians, or 3.14159 radians, is equal to the angle measuring 180.180. An angle measuring π2π2 radians, or 1.570796 radians, measures 90.90. To translate degrees to radians, we multiply the angle measure in degrees by π180.π180. For example, to write 4545 in radians, we have

    45(π180)=π4=0.785398radians.45(π180)=π4=0.785398radians.

    To translate radians to degrees, we multiply by 180π.180π. For example, to write 2π2π radians in degrees, we have

    2π(180π)=360.2π(180π)=360.

    Another example of translating radians to degrees and degrees to radians is 2π3.2π3. To write in degrees, we have 2π3(180π)=120.2π3(180π)=120. To write 3030 in radians, we have 30(π180)=π630(π180)=π6. However, we will use degrees throughout this chapter.

    FORMULA

    To translate an angle measured in degrees to radians, multiply by π180.π180.

    To translate an angle measured in radians to degrees, multiply by 180π.180π.

    Several angles are referred to so often that they have been given special names. A straight angle measures 180Figure 10.28.

    Four angles are depicted. Straight angle: 180 degrees. Right angle: 90 degrees. Acute angle: 60 degrees. Obtuse angle: 135 degrees.
    Figure 10.28 Classifying and Naming Angles

    An easy way to measure angles is with a protractor (Figure 10.29). A protractor is a very handy little tool, usually made of transparent plastic, like the one shown here.

    A protractor with its center labeled and an inch ruler is across the bottom.
    Figure 10.29 Protractor (credit: modification of work “School drawing tools” by Marco Verch/Flickr, CC BY 2.0)

    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.

    Checkpoint

    Make sure you correctly match the center mark of the protractor with the vertex of the angle to be measured. Otherwise, you will not get the correct measurement. Also, keep the protractor in a vertical position.

    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 Figure 10.30.

    Two rays, A C and A B make an acute angle.
    Figure 10.30 Naming an Angle

    We can name this angle BACBAC, or CABCAB, or A.A.

    Example 10.7

    Classifying Angles

    Determine which angles are acute, right, obtuse, or straight on the graph (Figure 10.31). You may want to use a protractor for this one.

    Eight rays are graphed on a square grid. All the rays originate from the same point, O. The rays, O L and O E are horizontal. The ray, O H is vertical. The ray, O K makes an acute angle with O L. The ray, O J makes an acute angle with O K. The rays, O J and O H make an acute angle. The ray, O G makes an acute angle with O H. The ray, O F makes an acute angle with O G. The rays, O E and O F make an acute angle. The eighth ray is perpendicular to O F and O A.
    Figure 10.31
    Answer

    Acute angles measure less than 90.90. Obtuse angles measure between 9090 and 180.180. Right angles measure 90.90. Straight angles measure 180.180.
    EOFEOGFOGGOHFOHHOJHOKJOKKOLJOLEOFEOGFOGGOHFOHHOJHOKJOKKOLJOL EOJEOKFOJFOKFOLGOKGOLEOJEOKFOJFOKFOLGOKGOL EOHHOLGOJEOHHOLGOJ EOLEOL

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

    Your Turn 10.7

    1.
    Determine which angles are acute, obtuse, right, and straight in the graph.
    Seven rays are graphed on a squared grid. All the rays originate from the same point, O. The rays, O F and O A are horizontal. The ray, O D is vertical. The ray, O E makes an acute angle with O F. The ray, O E makes an acute angle with O D. The ray, O C makes an acute angle with O D. The ray, O B makes an acute angle with O C. The rays, O B, and O A make an acute angle. The seventh ray is perpendicular to O F and O A.
    Figure 10.32

    Adjacent Angles

    Two angles with the same starting point or vertex and one common side are called adjacent angles. In Figure 10.33, angle DBCDBC is adjacent to CBACBA. Notice that the way we designate an angle is with a point on each of its two sides and the vertex in the middle.

    Three rays, B A, B C, and B D originate from the same point, B. The rays, B A, and B C make an acute angle. The rays, B C, and B D make an acute angle. The angle, A B D is acute.
    Figure 10.33 Adjacent Angles

    Supplementary Angles

    Two angles are supplementary if the sum of their measures equals 180.Figure 10.34, we are given that mFBE=35,mFBE=35, so what is mABE?mABE? These are supplementary angles. Therefore, because mABF=180mABF=180, and as 18035=145,18035=145, we have mABE=145.mABE=145.

    Five rays originate from the same point, B. The rays, B F, and B A are horizontal. The ray, B D is vertical. The ray, B E lies between B F and B D and it makes an acute angle with each ray. The ray, B C lies between B D and B A, and it makes an acute angle with each ray.
    Figure 10.34 Supplementary Angles

    Example 10.8

    Solving for Angle Measurements and Supplementary Angles

    Solve for the angle measurements in Figure 10.35.

    A horizontal line with a ray originating from its center. The line makes an acute angle, 5 x plus 2 with the ray, and an obtuse angle, 32 x minus 7 with the ray.
    Figure 10.35
    Answer

    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 180.180. Therefore, the sum of the two angles equals 180.180.

    Step 2:

    ( 32 x 7 ) + ( 5 x + 2 ) = 180 37 x 5 = 180 37 x = 185 x = 5 ( 32 x 7 ) + ( 5 x + 2 ) = 180 37 x 5 = 180 37 x = 185 x = 5

    Step 3: Find the measure of each angle:

    32 x 7 = 32 ( 5 ) 7 = 153 5 x + 2 = 5 ( 5 ) + 2 = 27 32 x 7 = 32 ( 5 ) 7 = 153 5 x + 2 = 5 ( 5 ) + 2 = 27

    Step 4: We check: 153+27=180.153+27=180.

    Your Turn 10.8

    1.
    Solve for the angle measurements in the figure shown.
    A horizontal line with a ray originating from its center. The line makes an acute angle, 2 x plus 5 with the ray, and an obtuse angle, 5 x with the ray.
    Figure 10.36

    Complementary Angles

    Two angles are complementary if the sum of their measures equals 90.Figure 10.37, we have mABC=30,mABC=30, and mABD=90.mABD=90. What is the mCBD?mCBD? These are complementary angles. Therefore, because 9030=60,9030=60, the CBD=60°.CBD=60°.

    Two lines, A B and B D intersect each other forming a right angle. A ray, B C makes an acute angle, 6 x minus 5 with the line, B A. Another ray originating from B makes an acute angle, 9 x with the line, B D. This ray and B C make an acute angle of 20 degrees.
    Figure 10.37 Complementary Angles

    Example 10.9

    Solving for Angle Measurements and Complementary Angles

    Solve for the angle measurements in Figure 10.38.

    Two lines intersect each other forming a right angle. A ray makes an acute angle, 7 x minus 5 with the horizontal line. Another ray originating from the intersection point of the lines makes an acute angle, 9 x minus 5 with the vertical line. An acute angle of 4 x is formed by these two rays.
    Figure 10.38
    Answer

    We have that

    ( 9 x 5 ) + 4 x + ( 7 x 5 ) = 90 20 x = 100 x = 5 ( 9 x 5 ) + 4 x + ( 7 x 5 ) = 90 20 x = 100 x = 5

    Then, m(9x5)=40m(9x5)=40, m(4x)=20m(4x)=20, and m(7x5)=30.m(7x5)=30.

    Your Turn 10.9

    1.
    Find the measure of each angle in the illustration.
    Two lines intersect each other forming a right angle. A ray makes an acute angle, 7 x plus 2 with the horizontal line. Another ray originating from the intersection point of the lines makes an acute angle, 6 x with the vertical line. An acute angle of 9 x is formed by these two rays.
    Figure 10.39

    Vertical Angles

    When two lines intersect, the opposite angles are called vertical angles, and vertical angles have equal measure. For example, Figure 10.40 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.

    Two lines intersect each other. One set of opposite angles is shaded.
    Figure 10.40 Vertical Angles

    Example 10.10

    Calculating Vertical Angles

    In Figure 10.41, one angle measures 40.40. Find the measures of the remaining angles.

    Two lines intersect each other. One set of opposite angles is labeled 1 and 3. The other set of opposite angles is labeled 2 and 40 degrees.
    Figure 10.41
    Answer

    The 40-degree angle and 22 are vertical angles. Therefore, m2=40.m2=40.

    Notice that 22 and 11 are supplementary angles, meaning that the sum of m2m2 and m1m1 equals 180.180. Therefore, m1=18040=140m1=18040=140.

    Since 11 and 33 are vertical angles, then m3m3 equals 140.140.

    Your Turn 10.10

    1.
    Given the two intersecting lines in the figure shown and /**/m\measuredangle 2 = {67^ \circ },/**/ find the measure of the remaining angles.
    Two lines intersect each other. One set of opposite angles is labeled 1 and 3. The other set of opposite angles is labeled 4 and 67 degrees.
    Figure 10.42

    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 10.43. Angles 1, 2, 7, and 8 are called exterior angles, and angles 3, 4, 5, and 6 are called interior angles.

    Two parallel lines, l subscript 1 and l subscript 2 are intersected by a transversal. The transversal makes four angles numbered 1, 2, 3, and 4 with the line, l subscript 1. The transversal makes four angles numbered 5, 6, 7, and 8 with the line, l subscript 2. 1, 2, 7, and 8 are exterior angles. 3, 4, 5, and 6 are interior angles.
    Figure 10.43 Transversal

    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, 3Figure 10.44.

    Two parallel lines, l subscript 1 and l subscript 2 are intersected by a transversal. The transversal makes four angles numbered 1, 2, 3, and 4 with the line, l subscript 1. The transversal makes four angles numbered 5, 6, 7, and 8 with the line, l subscript 2. 1, 2, 7, and 8 are exterior angles. 3, 4, 5, and 6 are interior angles. The alternate interior angles, 3 and 6 are highlighted.
    Figure 10.44 Alternate Interior Angles

    Alternate Exterior Angles

    Alternate exterior angles are exterior angles on opposite sides of the transversal and have the same measure. For example, in Figure 10.45, 22 and 77 are alternate exterior angles and have equal measures; 11 and 88 are alternate exterior angles and have equal measures as well.

    Two parallel lines, l subscript 1 and l subscript 2 are intersected by a transversal. The transversal makes four angles numbered 1, 2, 3, and 4 with the line, l subscript 1. The transversal makes four angles numbered 5, 6, 7, and 8 with the line, l subscript 2. 1, 2, 7, and 8 are exterior angles. 3, 4, 5, and 6 are interior angles. The alternate exterior angles, 2 and 7 are highlighted.
    Figure 10.45 Alternate Exterior Angles

    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 10.46, 11 and 55 are corresponding angles and have equal measures; 33 and 77 are corresponding angles and have equal measures; 22 and 66 are corresponding angles and have equal measures; 44 and 88 are corresponding angles and have equal measures as well.

    Two parallel lines, l subscript 1 and l subscript 2 are intersected by a transversal. The transversal makes four angles numbered 1, 2, 3, and 4 with the line, l subscript 1. The transversal makes four angles numbered 5, 6, 7, and 8 with the line, l subscript 2. 1, 2, 7, and 8 are exterior angles. 3, 4, 5, and 6 are interior angles. The corresponding angles, 1 and 5 are highlighted.
    Figure 10.46 Corresponding Angles

    Example 10.11

    Evaluating Space

    You live on the corner of First Avenue and Linton Street. You want to plant a garden in the far corner of your property (Figure 10.47) 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 150.150. What is the measure of the angle that will border your garden?

    Two streets, First Avenue and Linton Street intersect each other. One set of opposite angles is unknown and 150 degrees. The other set of opposite angles shows the garden on the left and blank on the right.
    Figure 10.47
    Answer

    As the angle between Linton Street and First Avenue is 150,150, the supplementary angle is 30.30. Therefore, the garden will form a 3030 angle at the corner of your property.

    Your Turn 10.11

    1.
    Suppose you have a similar property to the one in Figure 10.53, but the angle that corresponds to the garden corner is /**/{50^ \circ }/**/. What is the measure between the two cross streets?

    Example 10.12

    Determining Angles Formed by a Transversal

    In Figure 10.48 given that angle 3 measures 40,40, find the measures of the remaining angles and give a reason for your solution.

    Two parallel lines, l subscript 1 and l subscript 2 are intersected by a transversal. The transversal makes four angles numbered 1, 2, 40 degrees, and 4 with the line, l subscript 1. The transversal makes four angles numbered 5, 6, 7, and 8 with the line, l subscript 2. 1, 2, 7, and 8 are exterior angles. 40 degrees, 4, 5, and 6 are interior angles. The corresponding angles, 1 and 5 are highlighted.
    Figure 10.48
    Answer

    m2=m3=40m2=m3=40 by vertical angles.

    3=m73=m7 by corresponding angles.

    m7=m6=40m7=m6=40 by vertical angles.

    m1=18040=140m1=18040=140 by supplementary angles.

    m4=m1=140m4=m1=140 by vertical angles.

    m8=m1=140m8=m1=140 by alternate exterior angles.

    m5=m8=140m5=m8=140 by vertical angles.

    Your Turn 10.12

    1.
    In the given figure if /**/m\measuredangle 1 = {120^ \circ }/**/, find the /**/m\measuredangle 5/**/, /**/m\measuredangle 4/**/, and /**/m\measuredangle 8/**/.
    Two parallel lines, l subscript 1 and l subscript 2 are intersected by a transversal. The transversal makes four angles numbered 1, 2, 3, and 4 with the line, l subscript 1. The transversal makes four angles numbered 5, 6, 7, and 8 with the line, l subscript 2. 1, 2, 7, and 8 are exterior angles. 3, 4, 5, and 6 are interior angles. The corresponding angles, 1 and 5 are highlighted.
    Figure 10.49

    Example 10.13

    Measuring Angles Formed by a Transversal

    In Figure 10.50 given that angle 2 measures 23,23, find the measure of the remaining angles and state the reason for your solution.

    Two parallel lines, l subscript 1 and l subscript 2 are intersected by a transversal. The transversal makes four angles numbered 1, 23 degrees, 3, and 4 with the line, l subscript 1. The transversal makes four angles numbered 5, 6, 7, and 8 with the line, l subscript 2. 1, 23 degrees, 7, and 8 are exterior angles. 3, 4, 5, and 6 are interior angles. The corresponding angles, 1 and 5 are highlighted.
    Figure 10.50
    Answer

    m2=m3=23m2=m3=23 by vertical angles, because 22 and 33 are the opposite angles formed by two intersecting lines.

    m1=157m1=157 by supplementary angles to m2m2 or m3.m3. We see that 11 and 22 form a straight angle as does 11 and 3.3. A straight angle measures 180,180, so 18023=157.18023=157.

    m4=m1=157m4=m1=157 by vertical angles, because 44 and 11 are the two opposite angles formed by two intersecting lines.

    m5=m1=157m5=m1=157 by corresponding angles because they are the same angle formed by the transversal crossing two parallel lines, one exterior and one interior.

    m8=m5=157m8=m5=157 by vertical angles because 88 and 55 are the two opposite angles formed by two intersecting lines.

    m7=m2=23m7=m2=23 by alternate exterior angles because, like vertical angles, these angles are the opposite angles formed by the transversal intersecting two parallel lines.

    m6=m7=23m6=m7=23 by vertical angles because these are the opposite angles formed by two intersecting lines.

    Your Turn 10.13

    1.
    In the provided figure given that the /**/m\measuredangle 2 = {48^ \circ }/**/, find /**/m\measuredangle 1/**/, and /**/m\measuredangle 5./**/
    Two parallel lines, l subscript 1 and l subscript 2 are intersected by a transversal. The transversal makes four angles numbered 1, 2, 3, and 4 with the line, l subscript 1. The transversal makes four angles numbered 5, 6, 7, and 8 with the line, l subscript 2. 1, 2, 7, and 8 are exterior angles. 3, 4, 5, and 6 are interior angles. The corresponding angles, 1 and 5 are highlighted.
    Figure 10.51

    Example 10.14

    Finding Missing Angles

    Find the measures of the angles 1, 2, 4, 11, 12, and 14 in Figure 10.52 and the reason for your answer given that l1l1 and l2l2 are parallel.

    Two parallel lines, l subscript 1 and l subscript 2 are intersected by two transversals. The first transversal makes four angles numbered 62 degrees, 9, 7, and 8 with the line, l subscript 2. The second transversal makes four angles numbered 14, 62 degrees, 11, and 12 with the line, l subscript 2. The two transversals intersect at a point on the line, l subscript 1. Six angles are formed around the intersection point. The angles are labeled 1, 2, 62 degrees, 4, 5, and 6.
    Figure 10.52
    Answer

    m12=118m12=118, supplementary angles

    m14=118m14=118, vertical angles

    m11=62m11=62, vertical angles

    m4=62m4=62, corresponding angles

    m1=62m1=62, vertical angles

    m2=56m2=56, supplementary angles

    Your Turn 10.14

    1.
    Using Figure 10.58, find the measures of angles 5, 6, 7, 8, and 9.

    Who Knew?

    The Number 360

    Did you ever wonder why there are 360360 in a circle? Why not 100100 or 500?500? The number 360 was chosen by Babylonian astronomers before the ancient Greeks as the number to represent how many degrees in one complete rotation around a circle. It is said that they chose 360 for a couple of reasons: It is close to the number of days in a year, and 360 is divisible by 2, 3, 4, 5, 6, 8, 9, 10, …

    Check Your Understanding

    Classify the following angles as acute, right, obtuse, or straight.
    10.
    /**/m\measuredangle = {180^ \circ }/**/
    11.
    /**/m\measuredangle = {176^ \circ }/**/
    12.
    /**/m\measuredangle = {90^ \circ }/**/
    13.
    /**/m\measuredangle = {37^ \circ }/**/
    For the following exercises, determine the measure of the angles in the given figure.
    Two parallel lines, l subscript 1 and l subscript 2 are intersected by a transversal. The transversal makes four angles labeled 1, 31 degrees, 3, and 4 with the line, l subscript 1. The transversal makes four angles numbered 5, 6, 7, and 8 with the line, l subscript 2. 1, 31 degrees, 7, and 8 are exterior angles. 3, 4, 5, and 6 are interior angles.
    14.
    Find the measure of /**/\measuredangle 1/**/ and state the reason for your solution.
    15.
    Find the measure of /**/\measuredangle 3/**/ and state the reason for your solution.
    16.
    Find the measure of /**/\measuredangle 5/**/ and state the reason for your solution.

    Section 10.2 Exercises

    Classify the angles in the following exercises as acute, obtuse, right, or straight.
    1.
    /**/m\measuredangle = {32^ \circ }/**/
    2.
    /**/m\measuredangle = {120^ \circ }/**/
    3.
    /**/m\measuredangle = {180^ \circ }/**/
    4.
    /**/m\measuredangle = {90^ \circ }/**/
    5.
    /**/m\measuredangle = {110^ \circ }/**/
    6.
    /**/m\measuredangle = {45^ \circ }/**/
    7.
    Use the given figure to solve for the angle measurements.
    Two lines intersect each other forming a right angle. A ray originates from the intersection point of the lines. The ray makes an acute angle, 5 x plus 4 with the horizontal line. The ray makes an acute angle, 39 x minus 2 with the vertical line.
    8.
    Use the given figure to solve for the angle measurements.
    A horizontal line with two rays originating from its center. The first ray makes an angle, 3 x plus 2 with the horizontal axis. The angle formed between the two rays is labeled 3 x plus 7. The second ray makes an angle, 2 x plus 3 with the horizontal axis.
    9.
    Give the measure of the supplement to /**/{89^ \circ }./**/
    Use the given figure for the following exercises. Let angle 2 measure /**/{35^ \circ }/**/.
    Two parallel lines, l subscript 1 and l subscript 2 are intersected by a transversal. The transversal makes four angles numbered 1, 2, 3, and 4 with the line, l subscript 1. The transversal makes four angles numbered 5, 6, 7, and 8 with the line, l subscript 2. 1, 2, 7, and 8 are exterior angles. 3, 4, 5, and 6 are interior angles.
    10.
    Find the measure of angle 1 and state the reason for your solution.
    11.
    Find the measure of angle 3 and state the reason for your solution.
    12.
    Find the measure of angle 4 and state the reason for your solution.
    13.
    Find the measure of angle 5 and state the reason for your solution.
    14.
    Find the measure of angle 6 and state the reason and state the reason for your solution.
    15.
    Find the measure of angle 7 and state the reason for your solution.
    16.
    Find the measure of angle 8 and state the reason for your solution.
    17.
    Use the given figure to solve for the angle measurements.
    Two lines intersect each other. One set of opposite angles is labeled 5 x minus 129 and 2 x minus 21.
    Use the given figure for the following exercises.
    Two parallel lines are intersected by a transversal. The transversal makes four angles with the line, l subscript 1. Two angles are unknown. Two opposite angles are marked 3 and 50 degrees. The transversal makes four angles with the line, l subscript 2. 8 and 50 degrees are exterior angles. 3 is the interior angle.
    18.
    Find the measure of angle 3 and explain the reason for your solution.
    19.
    Find the measure of angle 8 and explain the reason for your solution.

    This page titled 10.3: Angles is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by OpenStax via source content that was edited to the style and standards of the LibreTexts platform.

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