CK-12 Geometry Read online

  The distance between the houses is about six miles.

  You can also use estimation to identify measurements in other geometric figures. Remember to include words like approximately, about, or estimation whenever you are finding an estimated answer.

  Ruler Postulate

  You have probably been using rulers to measure distances for a long time and you know that a ruler is a tool with measurement markings.

  Ruler Postulate: If you use a ruler to find the distance between two points, the distance will be the absolute value of the difference between the numbers shown on the ruler.

  The ruler postulate implies that you do not need to start measuring at the zero mark, as long as you use subtraction to find the distance. Note, we say “absolute value” here since distances in geometry must always be positive, and subtraction can yield a negative result.

  Example 2

  What distance is marked on the ruler in the diagram below? Assume that the scale is marked in centimeters.

  The way to use the ruler is to find the absolute value of difference between the numbers shown. The line segment spans from to .

  The absolute value of the difference between the two numbers shown on the ruler is . So, the line segment is long.

  Example 3

  Use a ruler to find the length of the line segment below.

  Line up the endpoints with numbers on your ruler and find the absolute value of the difference between those numbers. If you measure correctly, you will find that this segment measures or .

  Segment Addition Postulate

  Segment Addition Postulate: The measure of any line segment can be found by adding the measures of the smaller segments that comprise it.

  That may seem like a lot of confusing words, but the logic is quite simple. In the diagram below, if you add the lengths of and , you will have found the length of . In symbols, .

  Use the segment addition postulate to put distances together.

  Example 4

  The map below shows the distances between three collinear towns. Assume that the first third of the scale in black represents one inch.

  What is the distance between town and town ?

  You can see that the distance between town and town is eight miles. You can also see that the distance between town and town is five miles. Using the segment addition postulate, you can add these values together to find the total distance between town and town .

  The total distance between town and town is .

  Distances on a Grid

  In algebra you most likely worked with graphing lines in the coordinate grid. Sometimes you can find the distance between points on a coordinate grid using the values of the coordinates. If the two points line up horizontally, look at the change of value in the coordinates. If the two points line up vertically, look at the change of value in the coordinates. The change in value will show the distance between the points. Remember to use absolute value, just like you did with the ruler. Later you will learn how to calculate distance between points that do not line up horizontally or vertically.

  Example 5

  What is the distance between the two points shown below?

  The two points shown on the grid are at and . As these points line up vertically, look at the difference in the values.

  So, the distance between the two points is .

  Example 6

  What is the distance between the two points shown below?

  The two points shown on the grid are at and These points line up horizontally, so look at the difference in the values. Remember to take the absolute value of the difference between the values to find the distance.

  The distance between the two points is .

  Lesson Summary

  In this lesson, we explored segments and distances. Specifically, we have learned:

  How to measure distances using many different tools.

  To understand and apply the Ruler Postulate to measurement.

  To understand and apply the Segment Addition Postulate to measurement.

  How to use endpoints to identify distances on a coordinate grid.

  These skills are useful whenever performing measurements or calculations in diagrams. Make sure that you fully understand all concepts presented here before continuing in your study.

  Review Questions

  Use a ruler to measure the length of below.

  According to the ruler in the following image, how long is the cockroach?

  The ruler postulate says that we could have measured the cockroach in without using the marker as the starting point. If the same cockroach as the one in had its head at , where would its tail be on the ruler?

  Suppose is exactly in the middle of and . What is

  What is in the diagram below?

  Find in the diagram below:

  What is the length of the segment connecting and in the coordinate plane? Justify your answer.

  True or false: If and , then .

  True or false: .

  One of the statements in 8 or 9 is false. Show why it is false, and then change the statement to make it true.

  Review Answers

  Answers will vary depending on scaling when printed and the units you use.

  (yuck!).

  The tail would be at either or , depending on which way the cockroach was facing.

  .

  .

  .

  Since the points are at the same coordinate, we find the absolute value of the difference of the coordinates.

  False.

  True. , but the absolute value sign makes them both positive.

  Number 8 is false. See the diagram below for a counterexample. To make 8 true, we need to add something like: “If points , , and are collinear, and is between and , then if and , then .”

  Rays and Angles

  Learning Objectives

  Understand and identify rays.

  Understand and classify angles.

  Understand and apply the protractor postulate.

  Understand and apply the angle addition postulate.

  Introduction

  Now that you know about line segments and how to measure them, you can apply what you have learned to other geometric figures. This lesson deals with rays and angles, and you can apply much of what you have already learned. We will try to help you see the connections between the topics you study in this book instead of dealing with them in isolation. This will give you a more well-rounded understanding of geometry and make you a better problem solver.

  Rays

  A ray is a part of a line with exactly one endpoint that extends infinitely in one direction. Rays are named by their endpoint and a point on the ray.

  The ray above is called . The first letter in the ray’s name is always the endpoint of the ray, it doesn’t matter which direction the ray points.

  Rays can represent a number of different objects in the real world. For example, the beam of light extending from a flashlight that continues forever in one direction is a ray. The flashlight would be the endpoint of the ray, and the light continues as far as you can imagine so it is the infinitely long part of the ray. Are there other real-life objects that can be represented as rays?

  Example 1

  Which of the figures below shows

  A.

  B.

  C.

  D.

  Remember that a ray has one endpoint and extends infinitely in one direction. Choice A is a line segment since it has two endpoints. Choice B has one endpoint and extends infinitely in one direction, so it is a ray. Choice C has no endpoints and extends infinitely in two directions — it is a line. Choice D also shows a ray with endpoint Since we need to identify with endpoint , we know that choice B is correct.

  Example 2

  Use this space to draw .

  Remember that you are not expected to be an artist. In geometry, you simply need to draw figures that accurately represent the terms in question. This problem asks you to draw a ray. Begin with a line segment. Us
e your ruler to draw a straight line segment of any length.

  Now draw an endpoint on one end and an arrow on the other.

  Finally, label the endpoint and another point on the ray .

  The diagram above shows .

  Angles

  An angle is formed when two rays share a common endpoint. That common endpoint is called the vertex and the two rays are called the sides of the angle. In the diagram below, and form an angle, , or for short. The symbol is used for naming angles.

  The same basic definition for angle also holds when lines, segments, or rays intersect.

  Notation Notes:

  Angles can be named by a number, a single letter at the vertex, or by the three points that form the angle. When an angle is named with three letters, the middle letter will always be the vertex of the angle. In the diagram above, the angle can be written , or , or . You can use one letter to name this angle since point is the vertex and there is only one angle at point .

  If two or more angles share the same vertex, you MUST use three letters to name the angle. For example, in the image below it is unclear which angle is referred to by . To talk about the angle with one arc, you would write . For the angle with two arcs, you’d write .

  We use a ruler to measure segments by their length. But how do we measure an angle? The length of the sides does not change how wide an angle is “open.” Instead of using length, the size of an angle is measured by the amount of rotation from one side to another. By definition, a full turn is defined as degrees. Use the symbol for degrees. You may have heard “” used as slang for a “full circle” turn, and this expression comes from the fact that a full rotation is .

  The angle that is made by rotating through one-fourth of a full turn is very special. It measures and we call this a right angle. Right angles are easy to identify, as they look like the corners of most buildings, or a corner of a piece of paper.

  A right angle measures exactly .

  Right angles are usually marked with a small square. When two lines, two segments, or two rays intersect at a right angle, we say that they are perpendicular. The symbol is used for two perpendicular lines.

  An acute angle measures between and .

  An obtuse angle measures between and .

  A straight angle measures exactly . These are easy to spot since they look like straight lines.

  You can use this information to classify any angle you see.

  Example 3

  What is the name and classification of the angle below?

  Begin by naming this angle. It has three points labeled and the vertex is . So, the angle will be named or just . For the classification, compare the angle to a right angle. opens wider than a right angle, and less than a straight angle. So, it is obtuse.

  Example 4

  What term best describes the angle formed by Clinton and Reeve streets on the map below?

  The intersecting streets form a right angle. It is a square corner, so it measures .

  Protractor Postulate

  In the last lesson, you studied the ruler postulate. In this lesson, we’ll explore the Protractor Postulate. As you can guess, it is similar to the ruler postulate, but applied to angles instead of line segments. A protractor is a half-circle measuring device with angle measures marked for each degree. You measure angles with a protractor by lining up the vertex of the angle on the center of the protractor and then using the protractor postulate (see below). Be careful though, most protractors have two sets of measurements—one opening clockwise and one opening counterclockwise. Make sure you use the same scale when reading the measures of the angle.

  Protractor Postulate: For every angle there is a number between and that is the measure of the angle in degrees. You can use a protractor to measure an angle by aligning the center of the protractor on the vertex of the angle. The angle's measure is then the absolute value of the difference of the numbers shown on the protractor where the sides of the angle intersect the protractor.

  It is probably easier to understand this postulate by looking at an example. The basic idea is that you do not need to start measuring an angle at the zero mark, as long as you find the absolute value of the difference of the two measurements. Of course, starting with one side at zero is usually easier. Examples 5 and 6 show how to use a protractor to measure angles.

  Notation Note: When we talk about the measure of an angle, we use the symbols . So for example, if we used a protractor to measure in example 3 and we found that it measured , we could write .

  Example 5

  What is the measure of the angle shown below?

  This angle is lined up with a protractor at , so you can simply read the final number on the protractor itself. Remember you can check that you are using the correct scale by making sure your answer fits your angle. If the angle is acute, the measure of the angle should be less than . If it is obtuse, the measure will be greater than . In this case, the angle is acute, so its measure is .

  Example 6

  What is the measure of the angle shown below?

  This angle is not lined up with the zero mark on the protractor, so you will have to use subtraction to find its measure.

  Using the inner scale, we get .

  Using the outer scale, .

  Notice that it does not matter which scale you use. The measure of the angle is .

  Example 7

  Use a protractor to measure below.

  You can either line it up with zero, or line it up with another number and find the absolute value of the differences of the angle measures at the endpoints. Either way, the result is . The angle measures .

  Multimedia Link The following applet gives you practice measuring angles with a protractor Measuring Angles Applet.

  Angle Addition Postulate

  You have already encountered the ruler postulate and the protractor postulate. There is also a postulate about angles that is similar to the Segment Addition Postulate.

  Angle Addition Postulate: The measure of any angle can be found by adding the measures of the smaller angles that comprise it. In the diagram below, if you add and , you will have found .

  Use this postulate just as you did the segment addition postulate to identify the way different angles combine.

  Example 8

  What is in the diagram below?

  You can see that is . You can also see that is . Using the angle addition postulate, you can add these values together to find the total .

  So, is .

  Example 9

  What is in the diagram below given and ?

  To find , you must subtract from .

  So, .

  Lesson Summary

  In this lesson, we explored rays and angles. Specifically, we have learned:

  To understand and identify rays.

  To understand and classify angles.

  To understand and apply the Protractor Postulate.

  To understand and apply the Angle Addition Postulate.

  These skills are useful whenever studying rays and angles. Make sure that you fully understand all concepts presented here before continuing in your study.

  Review Questions

  Use this diagram for questions 1-4.

  Give two possible names for the ray in the diagram.

  Give four possible names for the line in the diagram.

  Name an acute angle in the diagram.

  Name an obtuse angle in the diagram.

  Name a straight angle in the diagram.

  Which angle can be named using only one letter?

  Explain why it is okay to name some angles with only one angle, but with other angles this is not okay.

  Use a protractor to find below:

  Given and , find .

  True or false: Adding two acute angles will result in an obtuse angle. If false, provide a counterexample.

  Review Answers

  or

  , , , or are four possible answers. There are more (how many?)

  or or

 
; Angle

  If there is more than one angle at a given vertex, then you must use three letters to name the angle. If there is only one angle at a vertex (as in angle above) then it is permissible to name the angle with one letter.

  .

  False. For a counterexample, suppose two acute angles measure and , then the sum of those angles is , but is still acute. See the diagram for a counterexample:

  Segments and Angles

  Learning Objectives

  Understand and identify congruent line segments.