1. Math and TERMINOLOGY for future Truckers
Unit 1: Geometry and Angle Measurements for Estimating Gradients
Created by Leecy Wise for Utah State University-Eastern Campus, Blanding, UT, 2014
© Utah State University-Eastern Campus: Blanding, 2014

2. When you are laughing, you are learning, so laugh, laugh, laugh!
Road humor
When you are laughing, you are learning, so laugh, laugh, laugh!

3. Why do truckers need math?
If you drive your own car or pick-up, you might want to use math in different ways. You might want to know what kind of mileage you are getting, or the number of miles you need to travel, or the amount of time it will take to get from one place to another. You might want to keep track of your expenses and create a budget for maintaining your vehicle.

4. A commercial trucker does not have those options
A commercial trucker does not have those options. A commercial trucker must use math for those reasons and many others. He must be able to handle financial transactions, complete log reports, measure loads, determine road characteristics, convert measurements, apply formulas, and much, much more. He must also be able to solve problems both on the job and on exams. To commercial truckers, math is a critical skill. It can be just as important sometimes as being able to drive in challenging weather.

5. It is important that you practice math skills using the resources in this series and those that your instructor will give you. Remember that practice makes perfect!

6. As you begin this or other units, always have a box or container filled with index cards. An index card box with alphabetical divisions is ideal. Add all new terms you learn to the box. For each new term you come across in every unit, write the term on one side of the card. On other side, write your own definition for the term, or simply draw a picture that will help you remember, or both! Then review your flashcards a lot.

7. In this unit, we will practice trucking math skills related to geometry.

8. Geometry for TRUCKERS ANDEQUIPMENT OPERATORS
Geometry is a math category of that deals with relationships between points, lines, surfaces, solids, and other dimensions. In other words, geometry is about shapes. The term, geometry, comes from Greek. (Get out your index card and add this term to your flashcard library.) “Geo” means “earth.” Metry comes from “measurement.” You might say that geometry studies measurements among earth’s objects. Truckers must know geometry well because they have to assess the degree of inclination on roads and equipment. They must be able to measure angles. Angles are at the heart of geometry. Add these terms to your flashcard collection.

9. angles
An angle measures the space between two intersecting lines or rays. In order to have an angle, two lines must intersect. They must meet at some point. That point between two intersecting lines forms an angle. Think about a road intersection. (Click to see animation.)
This is an angle

10. 1. RAys
A ray is an essential term when working with angles. Think of the sun. Sun rays start at the sun and go forever. Add this term to your flashcard collection. Imagine a line that starts at any point in your imagination and keeps going forever. That kind of line is called a RAY. Let’s call the starting point on a ray “the end point.” The sun would be called the end point for its rays. Of course, that doesn’t make much sense, does it? It should be the starting point, but it isn’t! The point at which you see the start of a ray is called the End Point!

11. To find out more about rays, go to..
Let’s label the end point on a ray as A. Now imagine that the ray continues, and you set additional points along that ray. You might label those as B, C, and so forth. It would look like the ray in the image below. (Click to see the image.) A B C
End Point
To find out more about rays, go to..

12. 2. lines
How do rays and lines relate to angles? When rays or lines intersect with each other, they form angles!!! They create a space that is framed by the two intersections. (Click to see the animation.)
These are lines that are intersecting each other.

13. Go back to your flashcard definition of an angle: an angle measures the space between two intersecting lines or rays at or close to the point where they meet. What is the difference between a line and a ray? Enter new terms to your library. In math, a LINE has no defined beginning or end point. Imagine that it just keeps going forever in both directions. A RAY, however does have a beginning point, called an “end point,” and then keeps going in one direction.

14. DRAWING LINES AND LABELING POINTS On a separate piece of paper, draw a long horizontal ray (a line with a “end point” that then goes forever. Now draw a line ( ) going vertically, and have it intersect the first ray you drew. Label that intersection as “A.” Repeat the process using a line going in a different direction and intersecting the first ray at a different point. Label that intersecting point as “B.” Continue the process drawing more lines, intersecting your first ray as before until you have points “C,” “D,” “E,” and “F.” Compare your image to what others drew. See sample images below of intersecting points.
Submit your image to your instructor A D C B F E

15. Look at the two moving lines below. Will they intersect at some point
Look at the two moving lines below. Will they intersect at some point? If you said, “Yes,” you are right. Now look at the two lines below. Will they ever intersect? Click to see. If you said, “No,” you would be right. Lines that run next to each other and never intersect are called parallel lines. Parallel lines do not form angles because they never intersect. Add that new term for your flashcard library.

16. 3. SEGMENTS
Get out that flashcard library again, and add another term to a card: SEGMENT. You will be working with segments to measure angles. A segment in geometry represents the distance between points on a a ray. The image below shows two segments along the ray. The segments are referred to as AB and BC. When you name a segment on a ray, you will often see a little arrow over the top to show it is a segment. Sometimes, you will find that each segment is identified by a lowercase letter instead. In the image below, the segments are name q and z instead of AB and CD. C B A
End Point
AB BC C B A
End Point q z

17. Decide True (T) or False (F). When you are done, click for answers.
This is a line .
This is a ray.
These lines intersect.
These lines form only three angles.
The yellow segment is called AB
“Geo” comes from Latin.
“Metry” means graphing.
The start of a ray is called and end point.
This road intersection sign forms four angles. T F T F T F T F T F A B C T F T F T F T F

18. Decide True (T) or False (F). When you are done, click for answers.
This is a line .
This is a ray.
These lines intersect.
These lines form only three angles.
The yellow segment is called AB
“Geo” comes from Latin.
“Metry” means graphing.
The start of a ray is called and end point.
This road intersection forms four angles. T F T F T F
No. They are parallel. T F
No. They form 4 angles. T F
No. It is BC. A B C T F
No. It comes from Greek. T F
No. It means measurement. T F T F
No. It forms two angles.

19. Now let’s talk more about angles
Now let’s talk more about angles. Go back to your flashcard definition of an angle: an angle measures the space between two intersecting lines/rays at or close to the point where they meet. Another way of saying is that an angle is made up from two intersecting rays that have the same end point. To represent an angle between two intersecting lines or rays, draw a curve that meets the two intersecting lines. Usually, that circle is small and close to the intersecting point.
This curved line represents an angle.

20. How many angles are formed in this image
How many angles are formed in this image? Work with a friend and compare your answers.

21. Angles are represented by curved lines that join the space between the two intersecting lines. Copy the following angles and draw curved lines to represent the space of the angles.

22. 4. MEASURING ANGLES
You know how to measure lengths and distances in the United States with units like inches, feet, yards, and miles. Most other countries measure distances in units like meters and kilometers. You know how to measure weights in the United States in units like pounds and tons. Most other countries measure weights in kilograms. Truckers everywhere must also know how to measure angles very well. Angles are measured in the same way all over the world. We measure angles using degrees in the same way that we use degrees to measure temperatures, but they mean different things.

23. Before you measure the gradient (the rise of fall) on earth’s surfaces, you must practice measuring angles using degrees. Truckers and heavy equipment operators must know how to work with angles to measure gradients. Those are the up and down parts of roads or the lift on equipment. Earlier you used curves to represent angles. That is because you were only showing a part of a whole circle. A curve is part of a circle.

24. Here’s the secret. Click to find out!
A full circle has 360 degrees, or 3600.
3600

25. Now that you know that a full circle has 3600, you can figure out how many degrees are in parts of circles! How many degrees are in half (1/2) a circle? Half of 360, of course, or How many degrees are in a quarter of one-fourth (1/4) of a circle? One fourth of 360! 360 ÷ 4 = 90, or And one sixth (1/6) would be half of that, or 450 .
1800
1800

26. You can now see how to measure angles since all angles are different degrees represented from a circle. The top and the bottom of a circle is the same. The bottom is simply a reflection of the top! We will measure angles here using only half a circle! Half a circle is 1800.

27. We know that if we draw and angle that represents ¼ of a circle, it will measure 900. Add 450 to that, and you will have = Draw an angle at half of 900, and you’ll have a 450 angle. Notice the different amounts of inclination that you see in each image.
900
1350
450

28. 450
900 00
1350

29. In order to talk more easily about angles, let’s give a few of them names. Add these to your flashcards. An angle that has rays going directly up and across, horizontally and vertically is called a RIGHT ANGLE. You will often see right angles shown with a square instead of a curve or ¼ circle.
900
Right angles are always right. They are never wrong!

30. An ACUTE ANGLE is an angle that is smaller than 900 .
Angles between these two blue lines are considered acute. Do you think they are cute?
450

31. An OBTUSE ANGLE is an angle that is greater than 900 .
1350
Look up the definition for the word “obtuse.” Do you think these angle are obtuse?

32. Click on the right angle.

33. Sorry. That was not the correct choice. Try again!

34. Good Job!

35. Click on any obtuse angle.

36. Sorry. That was not the correct choice. Try again!

37. Good Job!

38. Click on any acute angle.

39. Sorry. That was not the correct choice. Try again!

40. Good Job!

41. Go to the following site and play with the tools you see as you scroll down the page, then return here for more learning!

42. Now you are ready to measure angles
Now you are ready to measure angles. It is easy to see if an angle is a right angle (900) or a zero degree (00) angle. All angles in between are impossible to measure precisely just by looking at the proximity of the rays. In fact, on an exam, never assume you know how many degrees the images represent. Always solve problems using the number of degrees that they state on the test. Eyes are deceiving sometimes. You need a tool!
900 00

43. 5. PROTRACTORS
Take out your flashcard library, and add this new term to it. A protractor is a tool that measures the degree of any angle… usually on paper. Go to the next slide to watch a video on how to use a protractor to measure angles. Allow a few seconds for the video to load.

44. Allow a few seconds for the video to load.
Allow a few seconds for the video to load.

45. Now we are ready to talk more about how triangles help you measure and interpret the gradient amounts you will face as you work with trucks or heavy equipment.

46. 6. triangles
The images we used to show angles had only one angle each. There were only two intersecting rays or lines in each image. That means that they form one angle where they meet.
What happens when you draw an additional line to intercept both of those rays? How many angles are formed?
If you counted three, you are right! Draw the images above and draw curves to show the angles in each one.
“TRI” comes from both Greek and Latin. TRI means three (3). A triangle is a form that has three angles: tri-angle.
Make a list of all of the words you know that use “tri” prefix (word beginning), like tripod (3 legged), or triathlon (3-part competition).

47. Truckers need to know how to measure triangles. Why
Truckers need to know how to measure triangles. Why? Because road inclines or the inclination of an arm on a piece of heavy equipment are all measured using formulas relating to triangles. Knowing those formulas will avoid fatal accidents! You will learn to measure and estimate gradients in another unit, after you are very familiar with measuring angles.

48. As you can easily see, triangles have one side that represents the height. The triangle below has a right triangle with a height on the right side. You will need that measurement in measuring gradients when you work in trucking or heavy equipment.
HEIGHT

49. Whether you are working with obtuse, straight, or right triangles, it is helpful to remember that all triangles measure a total of The sum of all angles in a triangle is always 1800 because we are working with only half of a circle, or Reflect on why that is true and discuss it with your instructor or other friends.
1200
400
200
600
900
450
700
800
300

50. To get started measuring different aspect represented in triangles, let’s label a few parts. Earlier in this unit you learned that segments can be identified through labels. We use letters in caps to show the end points of a segment. We often use small-case letters to identify the segment itself. Let’s label a triangle that way. C B A
End Point
AB BC C B A
End Point q z

51. The triangle above has 3 intersecting points: A, B, and C.
HEIGHT A B C d h l
The triangle above has 3 intersecting points: A, B, and C.
It has three sides, or segments: d, l, and h. You will often find those sides represented with lines drawn over the two end points. For example:
Side d goes from A to B, also shown as AB
Side h goes from B to C, also shown as BC
Side l goes from A to C, also shown as CD.

52. Just for fun, let’s work with the sides of the following triangle.
A perimeter is the sum of the sides of an object. What is the perimeter of this triangle? ___________
What is the sum of d and l ? __________
What is the measurement of AB? _________
What is the measurement of angle “a”? __________
Click to check your answers. A B C d h l
13 cm
11 cm
8 cm a

53. What is the sum of d and l ? 13 + 11 = 24 cm B C d h l
13 cm
11 cm
8 cm a
A perimeter is the sum of the sides of an object. What is the perimeter of this triangle? = 32 cm
What is the sum of d and l ? = 24 cm
What is the measurement of AB? 13 cm
What is the measurement of angle “a”? 900 – It is a right angle.

54. Review and Practice
Complete the following worksheets and turn them into your instructor.
Use the symbol to identify or label angles.
To label segments or sides of angles, use a straight line above the letters, as you practiced earlier.
Click on each link below to access the worksheet. Check your answers only after you complete the worksheet.
Angles Basics Answer Sheet
Using the Protractor Answer Sheet
Find the Missing Angle Answer Sheet
Name the Angles in Figures Answer Sheet
AB DF

55. In the following unit on measuring slopes or gradients, you will learn new terms applying to angles. You will also be working with measurements relating to right triangles. In the next unit, you will practice describing the inclination or pitch of a slope or gradient in different ways. You are now ready to drive to your next unit: Slopes and Gradients in Trucking and Heavy Equipment.
8 miles
450

56. CONGRATULATIONS!