Geometry (A): READ: Angles Reading

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 $\overrightarrow{AB}$ . 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 $\overrightarrow{GH}?$

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 $H.$ Since we need to identify $\overrightarrow{GH}$ with endpoint $G$ , we know that choice B is correct.

Example 2

Use this space to draw $\overrightarrow{RT}$ .

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 $R$ and another point on the ray $T$ .

The diagram above shows $\overrightarrow{RT}$ .

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, $\overrightarrow{AB}$ and $\overrightarrow{AT}$ form an angle, $\angle BAT$ , or $\angle A$ for short. The symbol $\angle$ 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 $\angle{BAT}$ , or $\angle{TAB}$ , or $\angle{A}$ . You can use one letter to name this angle since point $A$ is the vertex and there is only one angle at point $A$ .
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 $\angle{L}$ . To talk about the angle with one arc, you would write $\angle{KLJ}$ . For the angle with two arcs, you’d write $\angle{JLM}$ .

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 $360$ degrees. Use the symbol $^\circ$ for degrees. You may have heard “ $360$ ” used as slang for a “full circle” turn, and this expression comes from the fact that a full rotation is $360^\circ$ .

The angle that is made by rotating through one-fourth of a full turn is very special. It measures $\frac{1} {4} \times 360^\circ = 90^\circ$ 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 $90^\circ$ .

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 $\bot$ is used for two perpendicular lines.

An acute angle measures between $0^\circ$ and $90^\circ$ .

An obtuse angle measures between $90^\circ$ and $180^\circ$ .

A straight angle measures exactly $180^\circ$ . 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 $U$ . So, the angle will be named $\angle{TUV}$ or just $\angle{U}$ . For the classification, compare the angle to a right angle. $\angle{TUV}$ 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 $90^\circ$ .

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 $0$ and $180$ 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 $m\angle$ . So for example, if we used a protractor to measure $\angle{TUV}$ in example 3 and we found that it measured $120^\circ$ , we could write $m \angle{TUV} = 120^\circ$ .

Example 5

What is the measure of the angle shown below?

This angle is lined up with a protractor at $0^\circ$ , 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 $90^\circ$ . If it is obtuse, the measure will be greater than $90^\circ$ . In this case, the angle is acute, so its measure is $50^\circ$ .

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 $|140-15|=|125|=125^{\circ}$ .

Using the outer scale, $|40-165|=|-125|=125^{\circ}$ .

Notice that it does not matter which scale you use. The measure of the angle is $125^\circ$ .

Example 7

Use a protractor to measure $\angle{RST}$ 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 $100^\circ$ . The angle measures $100^\circ$ .

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 $m\angle{ABC}$ and $m\angle{CBD}$ , you will have found $m\angle{ABD}$ .

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

Example 8

What is $m\angle{QRT}$ in the diagram below?

You can see that $m\angle{QRS}$ is $15^\circ$ . You can also see that $m\angle{SRT}$ is $30^\circ$ . Using the angle addition postulate, you can add these values together to find the total $m\angle{QRT}$ .

$15+30=45$

So, $m\angle{QRT}$ is $45^\circ$ .

Example 9

What is $m\angle{LMN}$ in the diagram below given $m\angle{LMO} = 85^\circ$ and $m\angle{NMO} = 53^\circ$ ?

To find $m\angle{LMN}$ , you must subtract $m\angle{NMO}$ from $m\angle{LMO}$ .

$85-53=32$

So, $m\angle{LMN} = 32^{\circ}$ .

Congruent Angles

You already know that congruent line segments have exactly the same length. You can also apply the concept of congruence to other geometric figures. When angles are congruent, they have exactly the same measure. They may point in different directions, have different side lengths, have different names or other attributes, but their measures will be equal.

Notation Notes:

When writing that two angles are congruent, we use the congruent symbol: $\angle{ABC} \cong \angle{ZYX}$ . Alternatively, the symbol $m\angle{ABC}$ refers to the measure of $\angle{ABC}$ , so we could write $m\angle{ABC} = m \angle{ZYX}$ and that has the same meaning as $\angle{ABC} \cong \angle{ZYX}$ . You may notice then, that numbers (such as measurements) are equal while objects (such as angles and segments) are congruent.
When drawing congruent angles, you use an arc in the middle of the angle to show that two angles are congruent. If two different pairs of angles are congruent, use one set of arcs for one pair, then two for the next pair and so on.

Use algebra to find a way to solve the problem below using this information.

Example 4

The two angles shown below are congruent.

What is the measure of each angle?

This problem combines issues of both algebra and geometry, so make sure you set up the problem correctly. It is given that the two angles are congruent, so they must have the same measurements. Therefore, you can set up an equation in which the expressions representing the angle measures are equal to each other.

$5x+7=3x+23$

Now that you have an equation with one variable, you can solve for the value of $x$ .

$5x+7 & = 3x+23\\ 5x-3x & = 23-7\\ 2x & = 16\\ x & = 8$

So, the value of $x$ is $8$ . You are not done, however. Use this value of $x$ to find the measure of one of the angles in the problem.

$m\angle{ABC} &=5x+7\\ &=5(8)+7\\ &=40+7\\ &=47$

Finally, we know $m\angle{ABC}=m\angle{XYZ}$ , so both of the angles measure $47^\circ$ .

Angle Bisectors

If a segment bisector divides a segment into two congruent parts, you can probably guess what an angle bisector is. An angle bisector divides an angle into two congruent angles, each having a measure exactly half of the original angle.

Angle Bisector Postulate: Every angle has exactly one bisector.

Example 5

The angle below measures $136^\circ$ .

If a bisector is drawn in this angle, what will be the measure of the new angles formed?

This is similar to the problem about the midpoint between the two houses. To find the measurements of the smaller angles once a bisector is drawn, divide the original angle measure by $2$ :

$136 \div 2 = 68$

So, each of the newly formed angles would measure $68^\circ$ when the $136^\circ$ angle is bisected.

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 $m\angle{PQR}$ below:
Given $m \angle{FNI} = 125^\circ$ and $m \angle{HNI} = 50^\circ$ , find $m \angle{FNH}$ .
True or false: Adding two acute angles will result in an obtuse angle. If false, provide a counterexample.

Review Answers

$CD$ or $CE$
$BD$ , $DB$ , $AB$ , or $BA$ are four possible answers. There are more (how many?)
$BDC$
$BDE$ or $BCD$ or $CDA$
$BDA$
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.
$|(50-130)| = |(-80)| = 80.$
$m\angle{FNH} = |125-50| = |75|=75^{\circ}$ .
False. For a counterexample, suppose two acute angles measure $30^\circ$ and $45^\circ$ , then the sum of those angles is $75^\circ$ , but $75^\circ$ is still acute. See the diagram for a counterexample:

Last modified: Friday, May 14, 2010, 9:23 AM