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Table of contents
  1. Preface
  2. 1 Sets
    1. Introduction
    2. 1.1 Basic Set Concepts
    3. 1.2 Subsets
    4. 1.3 Understanding Venn Diagrams
    5. 1.4 Set Operations with Two Sets
    6. 1.5 Set Operations with Three Sets
    7. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  3. 2 Logic
    1. Introduction
    2. 2.1 Statements and Quantifiers
    3. 2.2 Compound Statements
    4. 2.3 Constructing Truth Tables
    5. 2.4 Truth Tables for the Conditional and Biconditional
    6. 2.5 Equivalent Statements
    7. 2.6 De Morgan’s Laws
    8. 2.7 Logical Arguments
    9. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Projects
      5. Chapter Review
      6. Chapter Test
  4. 3 Real Number Systems and Number Theory
    1. Introduction
    2. 3.1 Prime and Composite Numbers
    3. 3.2 The Integers
    4. 3.3 Order of Operations
    5. 3.4 Rational Numbers
    6. 3.5 Irrational Numbers
    7. 3.6 Real Numbers
    8. 3.7 Clock Arithmetic
    9. 3.8 Exponents
    10. 3.9 Scientific Notation
    11. 3.10 Arithmetic Sequences
    12. 3.11 Geometric Sequences
    13. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  5. 4 Number Representation and Calculation
    1. Introduction
    2. 4.1 Hindu-Arabic Positional System
    3. 4.2 Early Numeration Systems
    4. 4.3 Converting with Base Systems
    5. 4.4 Addition and Subtraction in Base Systems
    6. 4.5 Multiplication and Division in Base Systems
    7. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Projects
      5. Chapter Review
      6. Chapter Test
  6. 5 Algebra
    1. Introduction
    2. 5.1 Algebraic Expressions
    3. 5.2 Linear Equations in One Variable with Applications
    4. 5.3 Linear Inequalities in One Variable with Applications
    5. 5.4 Ratios and Proportions
    6. 5.5 Graphing Linear Equations and Inequalities
    7. 5.6 Quadratic Equations with Two Variables with Applications
    8. 5.7 Functions
    9. 5.8 Graphing Functions
    10. 5.9 Systems of Linear Equations in Two Variables
    11. 5.10 Systems of Linear Inequalities in Two Variables
    12. 5.11 Linear Programming
    13. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  7. 6 Money Management
    1. Introduction
    2. 6.1 Understanding Percent
    3. 6.2 Discounts, Markups, and Sales Tax
    4. 6.3 Simple Interest
    5. 6.4 Compound Interest
    6. 6.5 Making a Personal Budget
    7. 6.6 Methods of Savings
    8. 6.7 Investments
    9. 6.8 The Basics of Loans
    10. 6.9 Understanding Student Loans
    11. 6.10 Credit Cards
    12. 6.11 Buying or Leasing a Car
    13. 6.12 Renting and Homeownership
    14. 6.13 Income Tax
    15. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  8. 7 Probability
    1. Introduction
    2. 7.1 The Multiplication Rule for Counting
    3. 7.2 Permutations
    4. 7.3 Combinations
    5. 7.4 Tree Diagrams, Tables, and Outcomes
    6. 7.5 Basic Concepts of Probability
    7. 7.6 Probability with Permutations and Combinations
    8. 7.7 What Are the Odds?
    9. 7.8 The Addition Rule for Probability
    10. 7.9 Conditional Probability and the Multiplication Rule
    11. 7.10 The Binomial Distribution
    12. 7.11 Expected Value
    13. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Formula Review
      4. Projects
      5. Chapter Review
      6. Chapter Test
  9. 8 Statistics
    1. Introduction
    2. 8.1 Gathering and Organizing Data
    3. 8.2 Visualizing Data
    4. 8.3 Mean, Median and Mode
    5. 8.4 Range and Standard Deviation
    6. 8.5 Percentiles
    7. 8.6 The Normal Distribution
    8. 8.7 Applications of the Normal Distribution
    9. 8.8 Scatter Plots, Correlation, and Regression Lines
    10. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  10. 9 Metric Measurement
    1. Introduction
    2. 9.1 The Metric System
    3. 9.2 Measuring Area
    4. 9.3 Measuring Volume
    5. 9.4 Measuring Weight
    6. 9.5 Measuring Temperature
    7. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  11. 10 Geometry
    1. Introduction
    2. 10.1 Points, Lines, and Planes
    3. 10.2 Angles
    4. 10.3 Triangles
    5. 10.4 Polygons, Perimeter, and Circumference
    6. 10.5 Tessellations
    7. 10.6 Area
    8. 10.7 Volume and Surface Area
    9. 10.8 Right Triangle Trigonometry
    10. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  12. 11 Voting and Apportionment
    1. Introduction
    2. 11.1 Voting Methods
    3. 11.2 Fairness in Voting Methods
    4. 11.3 Standard Divisors, Standard Quotas, and the Apportionment Problem
    5. 11.4 Apportionment Methods
    6. 11.5 Fairness in Apportionment Methods
    7. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  13. 12 Graph Theory
    1. Introduction
    2. 12.1 Graph Basics
    3. 12.2 Graph Structures
    4. 12.3 Comparing Graphs
    5. 12.4 Navigating Graphs
    6. 12.5 Euler Circuits
    7. 12.6 Euler Trails
    8. 12.7 Hamilton Cycles
    9. 12.8 Hamilton Paths
    10. 12.9 Traveling Salesperson Problem
    11. 12.10 Trees
    12. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  14. 13 Math and...
    1. Introduction
    2. 13.1 Math and Art
    3. 13.2 Math and the Environment
    4. 13.3 Math and Medicine
    5. 13.4 Math and Music
    6. 13.5 Math and Sports
    7. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Formula Review
      4. Projects
      5. Chapter Review
      6. Chapter Test
  15. A | Co-Req Appendix: Integer Powers of 10
  16. Answer Key
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
    7. Chapter 7
    8. Chapter 8
    9. Chapter 9
    10. Chapter 10
    11. Chapter 11
    12. Chapter 12
    13. Chapter 13
  17. Index
The exterior view of an architectural building.
Figure 10.21 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 10.22, 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 10.22 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 180180; a right angle measures 90;90; an acute angle is any angle whose measure is less than 90;90; and an obtuse angle is any angle whose measure is between 9090 and 180.180. See Figure 10.23.

Four angles are depicted. Straight angle: 180 degrees. Right angle: 90 degrees. Acute angle: 60 degrees. Obtuse angle: 135 degrees.
Figure 10.23 Classifying and Naming Angles

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.

A protractor with its center labeled and an inch ruler is across the bottom.
Figure 10.24 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 or before the points. When we are referring to the measure of the angle, we use the symbol mm. See Figure 10.25.

Two rays, A C and A B make an acute angle.
Figure 10.25 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 10.26). 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 10.26

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.

Adjacent Angles

Two angles with the same starting point or vertex and one common side are called adjacent angles. In Figure 10.27, 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 10.27 Adjacent Angles

Supplementary Angles

Two angles are supplementary if the sum of their measures equals 180.180. In Figure 10.28, 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 10.28 Supplementary Angles

Example 10.8

Solving for Angle Measurements and Supplementary Angles

Solve for the angle measurements in Figure 10.29.

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 10.29

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.

Complementary Angles

Two angles are complementary if the sum of their measures equals 90.90. In Figure 10.30, 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 10.30 Complementary Angles

Example 10.9

Solving for Angle Measurements and Complementary Angles

Solve for the angle measurements in Figure 10.31.

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 10.31

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.

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.

Two lines intersect each other. One set of opposite angles is shaded.
Figure 10.32 Vertical Angles

Example 10.10

Calculating Vertical Angles

In Figure 10.33, 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 10.33

Your Turn 10.10

1.
Given the two intersecting lines in the figure shown and m 2 = 67 , 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.

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.

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 10.34 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, 33 and 66 are alternate interior angles and have equal measure; 44 and 55 are alternate interior angles and have equal measure as well. See Figure 10.35.

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 10.35 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 10.36, 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 10.36 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 10.37, 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 10.37 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 10.38) 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 10.38

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 . What is the measure between the two cross streets?

Example 10.12

Determining Angles Formed by a Transversal

In Figure 10.39 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 10.39

Your Turn 10.12

1.
In the given figure if m 1 = 120 , find the m 5 , m 4 , and m 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.

Example 10.13

Measuring Angles Formed by a Transversal

In Figure 10.40 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 10.40

Your Turn 10.13

1.
In the provided figure given that the m 2 = 48 , find m 1 , and m 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.

Example 10.14

Finding Missing Angles

Find the measures of the angles 1, 2, 4, 11, 12, and 14 in Figure 10.41 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 10.41

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 = 180
11.
m = 176
12.
m = 90
13.
m = 37
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 1 and state the reason for your solution.
15.
Find the measure of 3 and state the reason for your solution.
16.
Find the measure of 5 and state the reason for your solution.
Classify the angles in the following exercises as acute, obtuse, right, or straight.
1 .
m = 32
2 .
m = 120
3 .
m = 180
4 .
m = 90
5 .
m = 110
6 .
m = 45
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 .
Use the given figure for the following exercises. Let angle 2 measure 35 .
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.
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