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 wellrounded 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
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 reallife 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
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. Use 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
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,
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
∠BAT , or∠TAB , or∠A . You can use one letter to name this angle since pointA is the vertex and there is only one angle at pointA .  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
∠L . To talk about the angle with one arc, you would write∠KLJ . For the angle with two arcs, you’d write∠JLM .
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
The angle that is made by rotating through onefourth of a full turn is very special. It measures
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
An acute angle measures between
An obtuse angle measures between
A straight angle measures exactly
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
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 halfcircle 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
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
Example 5
What is the measure of the angle shown below?
This angle is lined up with a protractor at
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
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
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
Use this postulate just as you did the segment addition postulate to identify the way different angles combine.
Example 8
What is
You can see that
So,
Example 9
What is
To find
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 14.
 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
m∠PQR below:  Given
m∠FNI=125∘ andm∠HNI=50∘ , findm∠FNH .  True or false: Adding two acute angles will result in an obtuse angle. If false, provide a counterexample.
Review Answers

CD orCE 
BD ,DB ,AB , orBA are four possible answers. There are more (how many?) 
BDC 
BDE orBCD orCDA 
BDA  Angle
C  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
C above) then it is permissible to name the angle with one letter. 
(50−130)=(−80)=80. 
m∠FNH=125−50=75=75∘ .  False. For a counterexample, suppose two acute angles measure
30∘ and45∘ , then the sum of those angles is75∘ , but75∘ is still acute. See the diagram for a counterexample:
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Date Created:
Feb 22, 2012Last Modified:
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