1. Angles
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Chapter 9
1.
Angles
This lesson introduces various types of angles in pre-algebra, such as obtuse, acute, right, and reflexive angles, while also explaining key concepts like congruent angles and parallel lines. It offers a clear overview of each angle's characteristics, providing foundational knowledge for students to understand geometry in practical scenarios.
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| | 26 Theory slides |
| | 13 Exercises - Grade E - A |
| | Each lesson is meant to take 1-2 classroom sessions |
Angles
Slide of 26
Angles are present everywhere. For example, when a person makes a peace sign ✌🏼 with their hand, the middle finger and index finger form an angle. Bricklayers 👷🏽♂️ always use angles at work. Different angles are formed at the intersections of roads. This lesson will focus on the essential concept of angles.
Catch-Up and Review
Here are a few recommended readings before getting started with this lesson.
Explore
Notebooks Open!
Think about a spiral notebook 📒. When it is opened, the two covers form an angle. As the cover is rotated, the angle measure changes.
If angles were classified by their measures, what names might they get?
Discussion
Angles
An angle is a plane figure formed by two rays that have the same starting point. This common point is called the vertex of the angle and the rays are the sides of the angle.
There are different ways to denote an angle and all involve the symbol ∠
in front of the name. One way is to name an angle by its vertex alone. Alternatively, it can be named by using all three points that make up the angle. In this case, the vertex is always in the middle of the name. Additionally, angles within a diagram can be denoted with numbers or lowercase Greek letters.
| Using the Vertex | Using the Vertex and One Point on Each Ray | Using a Number | Using Greek Letters |
|---|---|---|---|
| ∠B | ∠ABC or ∠CBA | ∠1 | ∠α or ∠β or ∠θ |
The measure of an angle, denoted by m∠, is the number of degrees between the rays. It is found by applying the Protractor Postulate.
Interior and Exterior of an Angle
An angle divides the plane into two parts.
- The region between the sides, or
interior
of the angle - The region outside the sides, or
exterior
of the angle
Discussion
Classifying an Angle From Its Measure
The measure of an angle can range from 0∘ to 360∘ or from 0 to 2π radians, a unit of measure that will be studied in the future. Angles can be classified according to their measures into six different types.
Concept
Acute Angle
An acute angle is an angle whose measure is greater than 0∘ but less than 90∘.
Discussion
Angles With a Measure of 90∘
Discussion
Angles With a Measure Ranging From 90∘ to 180∘
Discussion
Angles With a Measure of 180∘
As with right angles, the following type of angle involves only those angles whose measure is exactly 180∘.
Concept
Straight Angle
A straight angle is an angle whose measure is exactly 180∘.
Discussion
Angles With a Measure Greater Than 180∘
The fifth type of angle includes angles whose measure is greater than 180∘ but less than 360∘. This is the largest range of measures.
Concept
Reflexive Angle
A reflexive angle is an angle whose measure is greater than 180∘ but less than 360∘. An alternative name for this type of angle is reflex angle.
Discussion
Angles With a Measure of 360∘
Pop Quiz
Angles on a Clock Face
As time passes, the hands of a clock form different angles. Classify the indicated angle by estimating its measure.
Discussion
Angles With the Same Measure
When a laser is pointed at a mirror, the light beam is reflected in such a way that the angle between the incident beam and the mirror measures the same as the angle between the reflected beam and the mirror.
In the diagram, ∠1 and ∠3 have the same measure. Angles with the same measure have a special name.
Concept
Congruent Angles
Discussion
Classifying Angle Pairs Based on Their Position
Angles can also be classified based on their position relative to other angles.
Concept
Adjacent Angles
Going back to the diagram of the laser and the mirror, notice that the point where the beam hits the mirror is the vertex of three angles.
Discussion
Vertical Angles
Two angles are vertical angles if they are opposite angles formed by the intersection of two lines or line segments. In the diagram, vertical angles are marked with the same number of angle markers.
Vertical angles are always congruent.
Based on the characteristics of the diagram, the following relations hold true.
∠1≅∠3
∠2≅∠4
Example
Naming Pairs of Angles
In the diagram, lines AD and CE intersect at point X, and B is a point on the interior of ∠AXC.
a Name a pair of congruent angles, a pair of adjacent angles, and a pair of vertical angles, if there are any.
b Is there any relation between ∠BXC and ∠DXE?
c What is the measure of ∠EXA?
Answer
a
| Congruent Angles | Adjacent Angles | Vertical Angles |
|---|---|---|
| ∠AXB and ∠BXC ∠CXD and ∠EXA ∠DXE and ∠AXC |
∠AXB and ∠BXC ∠BXC and ∠CXD ∠CXD and ∠DXE ∠DXE and ∠EXA ∠EXA and ∠AXB |
∠CXD and ∠EXA ∠DXE and ∠AXC |
b There is no relation between ∠BXC and ∠DXE.
c m∠EXA=100∘
Hint
b Remove the unnecessary parts of the diagram to have a better look.
c Use the fact that ∠CXD measures 100∘.
Solution
a Start by identifying whether there are congruent angles, or angles with the same measure. In the diagram, only one angle measure is shown. However, congruent angles are also denoted with the same number of angle markers.
Since ∠AXB and ∠BXC have the same number of markers, the angles are congruent.
| Congruent Angles |
|---|
| ∠AXB and ∠BXC |
Next, focus on identifying adjacent angles. Adjacent angles have the same vertex, share one side, and they do not overlap. These three conditions are met by ∠DXE and ∠EXA.
In the diagram there are five pairs of adjacent angles.
| Adjacent Angles |
|---|
| ∠AXB and ∠BXC |
| ∠BXC and ∠CXD |
| ∠CXD and ∠DXE |
| ∠DXE and ∠EXA |
| ∠EXA and ∠AXB |
Finally, look for vertical angles. Vertical angles are opposite angles formed when two lines or line segments intersect. Since lines AD and CE intersect at X, they form two pairs of vertical angles. To make it easier to see, ignore the unnecessary parts of the diagram and focus on just these two lines.
From the diagram, ∠CXD and ∠EXA are vertical angles, as are ∠DXE and ∠AXC.
| Vertical Angles |
|---|
| ∠CXD and ∠EXA |
| ∠DXE and ∠AXC |
Because vertical angles are always congruent, the last pairs of angles are also congruent angles. All the information obtained from the diagram is summarized in the following table.
| Congruent Angles | Adjacent Angles | Vertical Angles |
|---|---|---|
| ∠AXB and ∠BXC ∠CXD and ∠EXA ∠DXE and ∠AXC |
∠AXB and ∠BXC ∠BXC and ∠CXD ∠CXD and ∠DXE ∠DXE and ∠EXA ∠EXA and ∠AXB |
∠CXD and ∠EXA ∠DXE and ∠AXC |
b To check whether ∠BXC and ∠DXE are related, ignore the irrelevant parts for a moment.
The angles have the same vertex but they do not have a common side. Therefore, they are not adjacent angles. Notice that ∠DXE has no angle marker and its measure seems to be greater than the measure of ∠BXC. Therefore, the angles are not congruent.
| ∠BXC and ∠DXE | ||
|---|---|---|
| Adjacent | Congruent | Vertical |
| × | × | ? |
Lastly, note that E, X, and C lie on the same line but D, X, and B do not. Therefore, ∠BXC and ∠DXE are not vertical angles. As such, there is no relation between these angles.
| ∠BXC and ∠DXE | ||
|---|---|---|
| Adjacent | Congruent | Vertical |
| × | × | × |
c From Part A, ∠EXA and ∠CXD are vertical angles, which also means they are congruent. This means that they have the same measure. Since the measure of ∠CXD is 100∘, the measure of ∠EXA is also 100∘.
Discussion
Angle Relationships Based on Measures
In addition to adjacent, vertical, and congruent angles, pairs of angles can be classified in three more ways based on the sum of their measures.
Concept
Complementary Angles
Two angles are complementary angles when the sum of their measures is 90∘.
In the diagram, ∠B and ∠E are complementary because the sum of their measures is equal to 90∘.
m∠B+m∠E=90∘
Notice that if two angles are complementary, they are by necessity acute angles. Also, if two complementary angles are adjacent, the angle formed by the not common sides is a right angle.
Example
Angle Formed by the Clock Hands
When the clock shows 09:00 and 55 seconds, the angle between the minute hand and the second hand is 30∘, while the angle between the minute hand and the hour hand is 90∘.
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Hint
The angle formed by the minute hand and the second hand is complementary to the angle formed by the second hand and the hour hand.
Solution
Start by marking a few points on the diagram to make it easy to reference its parts.
m∠JKL+m∠LKM=90∘
From the diagram, m∠JKL=30∘ and m∠LKM=5x∘. Substitute these values into the equation and solve for x.
Discussion
Supplementary Angles
Two angles are supplementary angles when the sum of their measures is 180∘.
In the diagram, ∠B and ∠E are supplementary because the sum of their measures equals 180∘.
m∠B+m∠E=180∘
If two angles are supplementary, either both are right angles or one is acute and the other obtuse. When two supplementary angles are adjacent, they are called a linear pair or straight angle pair. Notice that a linear pair forms a straight angle.
Example
Computer Screen Tilt
Some studies recommend tilting the computer screen 💻 slightly backwards between 10∘ and 20∘ for better posture and range of vision.
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Hint
The computer screen forms two supplementary angles with the table.
Solution
Notice that the computer screen forms two angles with the table and that these two angles together form a straight angle. This means that the angles are supplementary.
m∠PQR+m∠RQS=180∘
From the diagram, m∠PQR=110∘ and m∠RQS=(2x+30)∘. Substitute these measures into the equation and solve for x.
Discussion
Explementary Angles
Two angles are explementary angles, also called conjugate angles, when the sum of their measures is 360∘.
In the diagram, ∠B and ∠E are explementary because the sum of their measures is 360∘.
m∠B+m∠E=360∘
If two angles are explementary, either both are straight angles or one is a reflexive angle while the other can be acute, right, or obtuse. Notice that if two explementary angles are adjacent, they form a complete angle.
Example
Blind Spot Angle
When a human focuses their eyes 👀 on a point in front of them, their range of binocular vision is approximately 120∘.
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Hint
The binocular vision angle and the blind spot angle are explementary angles.
Solution
For simplicity, mark some points in the given diagram.
m∠ABC+m∠CBA=360∘
From the diagram, the binocular vision angle measures 120∘ and the blind spot angle measures (3x+15)∘. Substitute 120 for m∠ABC and (3x+15)∘ for m∠CBA and solve the equation for x.
If x=75, it can be also found that the blind spot angle measures 240∘.
Pop Quiz
Classifying Pairs of Angles
Classify each given pair of angles as complementary, supplementary, or explementary angles, or if they have no relationship.
Pop Quiz
Finding Angle Measures
For each of the given diagrams, find the value of x.
Discussion
Non-Intersecting Lines
Suppose two lines are drawn on a sheet of paper 📄. If the lines were to extend beyond the edges of the paper, there are only two possible cases for the lines: they either cross or they do not. When the lines do not intersect, they are called parallel lines.
Concept
Parallel Lines
Two coplanar lines — lines that are on the same plane — that do not intersect are said to be parallel lines. In a diagram, triangular hatch marks are drawn on lines to denote that they are parallel. The symbol ∥
is used to algebraically denote that two lines are parallel. In the diagram, lines m and ℓ are parallel.
Discussion
Special Case of Intersecting Lines
If two lines intersect each other and the angle between them is a right angle, the lines are called perpendicular lines.
Concept
Perpendicular Lines
Two coplanar lines — lines that are on the same plane — that intersect at a right angle are said to be perpendicular lines. The symbol ⊥
is used to algebraically denote that two lines are perpendicular. In the diagram, lines m and ℓ are perpendicular.
Pop Quiz
Classifying Pairs of Lines
For each pair of lines given, determine whether they are parallel, perpendicular, or neither. Remember, parallel lines are denoted with triangular hatch marks.
Closure
Drawing Three Lines
When three lines are drawn on a piece of paper 📄 and one of the lines cuts the other two, the paper is divided into six different regions. Additionally, eight angles are formed — four at each intersection point.
At each intersection point there are two pairs of vertical angles and four pairs of supplementary angles.
But this is not all! The angles formed at one intersection point can also be related to the angles formed at the other intersection point. These relationships will be explored later in the course.
| Vertical Angles | Supplementary Angles |
|---|---|
| ∠1 and ∠3 ∠2 and ∠4 |
∠1 and ∠2, ∠2 and ∠3 ∠3 and ∠4, ∠4 and ∠1 |
| ∠5 and ∠7 ∠8 and ∠9 |
∠5 and ∠6, ∠6 and ∠7 ∠7 and ∠8, ∠8 and ∠5 |
Angles
Exercises
Classify each of the following angles according to their measures.
m∠A=192∘
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m∠B=93∘
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m∠C=59∘
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m∠D=180∘
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There are six ways that angle can be classified according to its measure. Let's list them all in a table.
| Name | Measure |
|---|---|
| Acute | Greater than 0^(∘) but less than 90^(∘) |
| Right | Exactly 90^(∘) |
| Obtuse | Greater than 90^(∘) but less than 180^(∘) |
| Straight | Exactly 180^(∘) |
| Reflexive | Greater than 180^(∘) but less than 360^(∘) |
| Complete | Exactly 360^(∘) |
In our case, we are given that m∠ A=192^(∘). Therefore, we can conclude that ∠ A is a reflexive angle. Let's draw an example of this angle.
We are given that m∠ B=93^(∘). Notice that this measure is greater than 90^(∘) but less than 180^(∘). Using the table we wrote in Part A, we can state that ∠ B is an obtuse angle.
The measure of ∠ C is 59^(∘), which is less than 90^(∘). This means that it belongs to the first row of the table from Part A. For this reason, we can conclude that ∠ C is an acute angle.
Notice that the measure of ∠ D is exactly 180^(∘). According to the classification we wrote in Part A, we can say that ∠ D is a straight angle.
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For each of the following diagrams, identify the angle types.
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We start by remembering that right angles are denoted by square angle markers. For this reason, we can say that ∠ 3 is a right angle.
To classify ∠ 1 and ∠ 2, we can either estimate their measures or use a protractor and find their actual measures. In this case, let's go with the second option.
We got that m∠ 1=120^(∘) and m∠ 2 = 55^(∘). Since the measure of ∠ 1 is greater than 90^(∘) and less than 180^(∘), we conclude that it is obtuse. On the other hand, because the measure of ∠ 2 is less than 90^(∘), it is an acute angle. ∠ 1 &→ Obtuse ∠ 2 &→ Acute ∠ 3 &→ Right
Notice that ∠ 2 has a square angle marker, just like in Part A. Therefore, we can say that ∠ 2 is a right angle.
According to the diagram, ∠ 1 has a measure greater than 180^(∘) but less than 360^(∘). This means that ∠ 1 is a reflexive angle. On the other hand, the measure of ∠ 3 seems to be exactly 180^(∘), so it is a straight angle. Nevertheless, let's confirm these claims by measuring the angles with a protractor.
We got that m∠ 1= 180^(∘) + 120^(∘) = 300^(∘) and m∠ 3=180^(∘), which confirms our conclusions. ∠ 1 &→ Reflexive ∠ 2 &→ Right ∠ 3 &→ Straight
We again see that one of the angle markers is a square, which means that the angle measures 90^(∘). Therefore, we have that ∠ 1 is a right angle.
At first glance, it looks like ∠ 2 is acute and ∠ 3 is obtuse. To verify our suspicions, let's measure the angles with the help of a protractor.
We have that m∠ 2=37^(∘) and m∠ 3=127^(∘). Therefore, as expected, ∠ 2 is acute and ∠ 3 is obtuse. ∠ 1 &→ Right ∠ 2 &→ Acute ∠ 3 &→ Obtuse
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For each of the following diagrams, identify the characteristics that the labeled pair of angles meet.
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{"type":"multichoice","form":{"alts":["Congruent","Adjacent","Vertical","Complementary","Supplementary","Explementary"],"noSort":true},"formTextBefore":"","formTextAfter":"","answer":[0,2]}
{"type":"multichoice","form":{"alts":["Congruent","Adjacent","Vertical","Complementary","Supplementary","Explementary"],"noSort":true},"formTextBefore":"","formTextAfter":"","answer":[4]}
{"type":"multichoice","form":{"alts":["Congruent","Adjacent","Vertical","Complementary","Supplementary","Explementary"],"noSort":true},"formTextBefore":"","formTextAfter":"","answer":[0,1,3]}
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Let's begin by analyzing the positioning of the angles. We can see that they share a vertex, have a common side — the ray pointing going to the upper left — and do not overlap.
Consequently, we can say that the angles are adjacent. Because of this, the angles cannot be vertical angles. Next, let's consider their measures. The angle measures are different, which means they are not congruent. Finally, let's find the sum of their measures. 110^(∘) + 130^(∘) = 240^(∘) Since the sum is not 90^(∘), 180^(∘), or 360^(∘), we can conclude that the angles are not complementary, supplementary, or explementary. To finish, let's summarize all the characteristics the given pair of angles meet. lcl * Congruent & & * Complementary ✓ Adjacent & & * Supplementary * Vertical & & * Explementary
As we did in Part A, let's start by looking at the positioning of the angles. We can see that they are formed by the intersection of two lines and that they are opposite angles.
Therefore, the angles are vertical angles. For this reason, they cannot be adjacent. Let's continue by considering their measures. We can see that both angles have the same measure, so they are congruent. Next, let's find the sum of their measures. 44^(∘) + 44^(∘) = 88^(∘) The sum is not 90^(∘), 180^(∘), or 360^(∘), so the angles are not complementary, supplementary, or explementary. Finally, let's gather all the information we got from the given pair of angles. lcl ✓ Congruent & & * Complementary * Adjacent & & * Supplementary ✓ Vertical & & * Explementary
Once again, we start by looking at how the angles are placed.
This time the angles do not have any common vertices or sides, so they are neither adjacent nor vertical. Additionally, their measures are different, implying that they are not congruent either. What a start! lcl * Congruent * Adjacent * Vertical Next, let's find the sum of their measures. 110^(∘) + 70^(∘) = 180^(∘) Since the sum is 180^(∘), we can conclude that the angles are supplementary. This means that they are neither complementary nor explementary. Before moving on, let's write all the information about the angles. lcl * Congruent & & * Complementary * Adjacent & & ✓ Supplementary * Vertical & & * Explementary
We have just one more pair of angles to consider, so let's do it. We start by noticing that the angles share a vertex and a side and they do not overlap.
Therefore, the angles are adjacent, which also means that they cannot be vertical angles. Additionally, we see that both angles have the same measure, so they are congruent. Next, let's find their sum. 45^(∘) + 45^(∘) = 90^(∘) Since the sum is 90^(∘), the angles are complementary. This means that they are not supplementary or explementary. Finally, let's summarize all the information about the angles. lcl ✓ Congruent & & ✓ Complementary ✓ Adjacent & & * Supplementary * Vertical & & * Explementary
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For each of the following diagrams, identify the characteristics that ∠1 and ∠2 meet, if any.
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{"type":"multichoice","form":{"alts":["Congruent","Adjacent","Vertical","Complementary","Supplementary","Explementary"],"noSort":true},"formTextBefore":"","formTextAfter":"","answer":[0,3]}
{"type":"multichoice","form":{"alts":["Congruent","Adjacent","Vertical","Complementary","Supplementary","Explementary"],"noSort":true},"formTextBefore":"","formTextAfter":"","answer":[0,1,4]}
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Let's begin by noticing that the angles are formed by the intersection of two lines. However, the angles are not opposite one another, so they are not vertical angles. Instead, they share a vertex and one side, and they do not overlap. Therefore, the angles are adjacent.
Since the angles are denoted by a different number of angle markers, we can say that the angles are not congruent. Additionally, we can see that ∠ 1 and ∠ 2 together form a straight angle. This means that the sum of their measures is 180^(∘). Therefore, the angles are supplementary. m∠ 1 + m∠ 2 = 180^(∘) Because of this, the angles are neither complementary nor explementary. To finish, let's summarize all the characteristics the given pair of angles meet. lcl * Congruent & & * Complementary ✓ Adjacent & & ✓ Supplementary * Vertical & & * Explementary
The first thing we notice is that both angles are denoted using the same number of angle markers. This implies that the angles have the same measure — in other words, they are congruent.
We can see that the angles share a vertex, but they do not have a common side. This means that they are not adjacent. Also, they are not vertical either. Next, notice that ∠ 1 and ∠ 2 can combine with the right angle between them to form a straight angle with a measure of 180^(∘). Let's use this to find the sum of the measures of angles 1 and 2. m∠ 1 + 90^(∘) + m∠ 2 = 180^(∘) ⇓ m∠ 1 + m∠ 2 = 90^(∘) The sum of the measures of ∠ 1 and ∠ 2 is 90^(∘). This implies that the angles are complementary. If the angles are complementary, they cannot be supplementary or explementary. Before moving on, let's write all the information about the angles. lcl ✓ Congruent & & ✓ Complementary * Adjacent & & * Supplementary * Vertical & & * Explementary
Notice that the two angles are denoted by square angle markers. This means that both angles are right angles. Because of this, they are congruent. Additionally, the angles share a vertex and a side and do not overlap, so they are adjacent.
As we can see, the angles are not vertical. Finally, we see that both angles form a straight angle, which implies that they are supplementary. As a consequence, they are neither complementary nor explementary. Finally, let's gather all the information we got from the given pair of angles. lcl ✓ Congruent & & * Complementary ✓ Adjacent & & ✓ Supplementary * Vertical & & * Explementary
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Consider the following diagram where the rays that form a line are colored the same.
Fill in the blanks in the following sentences. Select all the characteristics that apply to the indicated angles!
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∠1 and ∠5 are . |
{"type":"multichoice","form":{"alts":["Congruent","Adjacent","Vertical","Complementary","Supplementary","Explementary","Not related"],"noSort":true},"formTextBefore":"","formTextAfter":"","answer":[0,2]}
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∠3 and ∠4 are . |
{"type":"multichoice","form":{"alts":["Congruent","Adjacent","Vertical","Complementary","Supplementary","Explementary","Not related"],"noSort":true},"formTextBefore":"","formTextAfter":"","answer":[1,3]}
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∠2 and ∠7 are . |
{"type":"multichoice","form":{"alts":["Congruent","Adjacent","Vertical","Complementary","Supplementary","Explementary","Not related"],"noSort":true},"formTextBefore":"","formTextAfter":"","answer":[6]}
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∠1 and ∠2 are . |
{"type":"multichoice","form":{"alts":["Congruent","Adjacent","Vertical","Complementary","Supplementary","Explementary","Not related"],"noSort":true},"formTextBefore":"","formTextAfter":"","answer":[1,3]}
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To determine the relationship between ∠ 1 and ∠ 5, let's focus our attention on them and ignore the other angles.
We can see that these two angles are formed by the intersection of two lines and are opposite one another. This means that they are vertical angles. For this reason, we can also say that the angles are congruent. ✓ & Congruent ✓ & Vertical Since the angles are vertical, they cannot be adjacent angles. Also, even though we do not know their measures, we can see that they are not complementary, supplementary, or explementary. This can be checked using a protractor. Finally, let's sum up all the characteristics. lcl ✓ Congruent & & * Complementary * Adjacent & & * Supplementary ✓ Vertical & & * Explementary
As before, let's focus our attention on the two angles in question, ∠ 3 and ∠ 4.
We can see that the angles have the same vertex, share a common side and do not overlap. This means that they are adjacent angles, which also means that they are not vertical angles. ✓ & Adjacent * & Vertical Even though the angle measures are not given, we can see that they have different measures, so they are not congruent. Let's now try to estimate the sum of their measures. To do so, let's look at ∠ 8, which is formed by extending the non-common sides of angles 3 and 4.
Notice that ∠ 8 is denoted by a square angle marker, which means that it is a right angle. Next, we see that ∠ 3 and ∠ 4 together form an angle that is opposite to ∠ 8 — in other words, ∠ 8 and the angle formed by angles 3 and 4 are vertical angles.
Because vertical angles are congruent, we can conclude that ∠ 3 and ∠ 4 together form a right angle and the sum of their measures is 90^(∘). This means that they are complementary angles. m∠ 3 + m∠ 4 = 90^(∘) ⇓ ✓ Complementary This last conclusion leads us to the conclusion that the angles are neither supplementary nor explementary. lcl * Congruent & & ✓ Complementary ✓ Adjacent & & * Supplementary * Vertical & & * Explementary
Now we want to see whether ∠ 2 and ∠ 7 are related. Let's start by focusing our attention on them and ignoring the rest of the angles.
The angles only share a vertex, so they are not adjacent. It looks like ∠ 2 has a greater measure than ∠ 7, so they are not congruent. We need to be careful because the angles might seem to be vertical, but notice that the pink and orange rays do not form a line. This fact means that ∠ 2 and ∠ 7 are not vertical angles.
Finally, although the measures are not given, it looks like the angles are not complementary, supplementary, or explementary. We can confirm this by using a protractor. Therefore, after the study we just did, we can conclude that the angles are not related. lcl * Congruent & & * Complementary * Adjacent & & * Supplementary * Vertical & & * Explementary
Finally, let's verify whether ∠ 1 and ∠ 2 are somehow related. As before, let's focus our attention on them.
We can see that the angles share a vertex, have a common side, and do not overlap. Therefore, they are adjacent angles. Because of this, they are not vertical angles. We can see that ∠ 2 has a greater measure, which means they are not congruent. * & Congruent ✓ & Adjacent * & Vertical As we did in Part B, let's try to estimate the sum of the measures of angles 1 and 2. If we extend the side of ∠ 2 that is not common to ∠ 1 beyond the vertex, we see that this new side along with one side of ∠ 1 form ∠ 8.
Notice that angles 1, 2, and 8 together form a straight angle with a sum of 180^(∘). m∠ 1 + m∠ 2 + m∠ 8 = 180^(∘) Because ∠ 8 is marked by a square angle marker, it is a right angle and has a measure of 90^(∘). Let's substitute this into the previous equation. m∠ 1 + m∠ 2 + 90^(∘) &= 180^(∘) &⇓ m∠ 1 + m∠ 2 &= 90^(∘) We got that the measures of angles 1 and 2 add up to 90^(∘). This implies that they are complementary angles. Due to this, they are neither supplementary nor explementary. Lastly, let's summarize all the characteristics of the given angles. lcl * Congruent & & ✓ Complementary ✓ Adjacent & & * Supplementary * Vertical & & * Explementary
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Consider the following diagram.
{"type":"choice","form":{"alts":["<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.69224em;vertical-align:0em;\"><\/span><span class=\"mord\">\u2220<\/span><span class=\"mord mathdefault\">A<\/span><span class=\"mord mathdefault\" style=\"margin-right:0.02778em;\">O<\/span><span class=\"mord mathdefault\" style=\"margin-right:0.05017em;\">B<\/span><\/span><\/span><\/span> and <span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.69224em;vertical-align:0em;\"><\/span><span class=\"mord\">\u2220<\/span><span class=\"mord mathdefault\" style=\"margin-right:0.02778em;\">D<\/span><span class=\"mord mathdefault\" style=\"margin-right:0.02778em;\">O<\/span><span class=\"mord mathdefault\" style=\"margin-right:0.05764em;\">E<\/span><\/span><\/span><\/span>","<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.69224em;vertical-align:0em;\"><\/span><span class=\"mord\">\u2220<\/span><span class=\"mord mathdefault\">A<\/span><span class=\"mord mathdefault\" style=\"margin-right:0.02778em;\">O<\/span><span class=\"mord mathdefault\" style=\"margin-right:0.05017em;\">B<\/span><\/span><\/span><\/span> and <span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.69224em;vertical-align:0em;\"><\/span><span class=\"mord\">\u2220<\/span><span class=\"mord mathdefault\" style=\"margin-right:0.05017em;\">B<\/span><span class=\"mord mathdefault\" style=\"margin-right:0.02778em;\">O<\/span><span class=\"mord mathdefault\" style=\"margin-right:0.07153em;\">C<\/span><\/span><\/span><\/span>","<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.69224em;vertical-align:0em;\"><\/span><span class=\"mord\">\u2220<\/span><span class=\"mord mathdefault\">A<\/span><span class=\"mord mathdefault\" style=\"margin-right:0.02778em;\">O<\/span><span class=\"mord mathdefault\" style=\"margin-right:0.05017em;\">B<\/span><\/span><\/span><\/span> and <span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.69224em;vertical-align:0em;\"><\/span><span class=\"mord\">\u2220<\/span><span class=\"mord mathdefault\" style=\"margin-right:0.13889em;\">F<\/span><span class=\"mord mathdefault\" style=\"margin-right:0.02778em;\">O<\/span><span class=\"mord mathdefault\">A<\/span><\/span><\/span><\/span>","<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.69224em;vertical-align:0em;\"><\/span><span class=\"mord\">\u2220<\/span><span class=\"mord mathdefault\" style=\"margin-right:0.02778em;\">D<\/span><span class=\"mord mathdefault\" style=\"margin-right:0.02778em;\">O<\/span><span class=\"mord mathdefault\" style=\"margin-right:0.05764em;\">E<\/span><\/span><\/span><\/span> and <span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.69224em;vertical-align:0em;\"><\/span><span class=\"mord\">\u2220<\/span><span class=\"mord mathdefault\" style=\"margin-right:0.13889em;\">F<\/span><span class=\"mord mathdefault\" style=\"margin-right:0.02778em;\">O<\/span><span class=\"mord mathdefault\">A<\/span><\/span><\/span><\/span>"],"noSort":false},"formTextBefore":"","formTextAfter":"","answer":0}
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To determine which pair of angles does not belong with the other three, we need to identify common characteristics between them. Let's analyze each pair of angles separately.
∠ AOB and ∠ BOC
To look at the angles AOB and BOC in detail, let's focus on them and ignore the other angles.
As we can see, the angles share the vertex O, have a common side OB, and do not overlap. Therefore, they are adjacent angles. This could be a common characteristic. However, there is another characteristic that these angles meet. To figure it out, let's add in OD.
From the diagram, the angles AOB, BOC, and COD together form a straight angle. This means that their measures add up to 180^(∘). m∠ AOB + m∠ BOC + m∠ COD = 180^(∘) Since ∠ COD is drawn with a square angle marker, it is a right angle and its measure is 90^(∘). Let's substitute it into the previous equation. m∠ AOB + m∠ BOC + 90^(∘) = 180^(∘) ⇓ m∠ AOB + m∠ BOC = 90^(∘) We found that the measures of ∠ AOB and ∠ BOC add up to 90^(∘). Therefore, they are complementary angles. There are two characteristics that this pair of angles meet.
∠ AOB and ∠ BOC are adjacent and complementary angles.
∠ AOB and ∠ FOA
Let's analyze ∠ AOB and ∠ FOA.
We can see that the angles share the vertex O, have a common side OA, and do not overlap. Therefore, they are adjacent angles, just like the first pair of angles! Let's determine whether this pair of angles are also complementary. To do so, let's add in OE.
From the diagram, angles EOF, FOA, and AOB together form a straight angle with a measure of 180^(∘). m∠ EOF + m∠ FOA + m∠ AOB = 180^(∘) Because ∠ EOF is drawn with a square angle marker, it is a right angle and measures 90^(∘). Let's substitute it into the previous equation. 90^(∘) + m∠ FOA + m∠ AOB = 180^(∘) ⇓ m∠ FOA + m∠ AOB = 90^(∘) The measures of ∠ FOA and ∠ AOB add up to 90^(∘). As such, they are complementary angles, just like the first pair!
∠ AOB and ∠ FOA are adjacent and complementary angles.
∠ DOE and ∠ FOA
Now let's consider ∠ DOE and ∠ FOA.
The angles share the vertex O and do not overlap, but they do not have a common side, meaning that they are not adjacent angles. This could be the pair of angles that does not belong to the group. Let's see if they are complementary. To do so, notice that the angle formed by OE and OF is a right angle.
We can see that angles DOE, EOF, and FOA together form a straight angle, which means that their measures add up to 180^(∘). m∠ DOE + m∠ EOF + m∠ FOA = 180^(∘) Because ∠ EOF is a right angle, its measure is 90^(∘). Let's substitute it into the previous equation. m∠ DOE + 90^(∘) + m∠ FOA = 180^(∘) ⇓ m∠ DOE + m∠ FOA = 90^(∘) The measures of ∠ DOE and ∠ FOA add up to 90^(∘). Therefore, the angles are complementary.
∠ DOE and ∠ FOA are complementary angles.
Although this pair of angles shares only one characteristic with the previous two pairs, we will not draw any conclusions until we consider the last pair.
∠ AOB and ∠ DOE
Lastly, let's analyze ∠ AOB and ∠ DOE.
We can see that the angles do not have a common side, which means that they are not adjacent, just like the previous pair. Now, let's determine whether the angles are complementary. To do so, start by noticing that the angles are vertical, which means that they are congruent. m∠ AOB = m∠ DOE For a moment, let's suppose that the angles were complementary. If that were true, the sum of their measures would be 90^(∘), which implies that each angle measures 45^(∘). However, remember that we already we concluded that ∠ AOB and ∠ BOC are complementary. m∠ AOB + m∠ BOC = 90^(∘) Let's substitute 45^(∘) for m∠ AOB. 45^(∘) + m∠ BOC = 90^(∘) ⇓ m∠ BOC = 45^(∘) As we can see, the measure of ∠ BOC would also be 45^(∘). But this is not possible because we can see that the measure of ∠ AOB is greater than the measure of ∠ BOC. This contradiction comes from supposing that ∠ AOB and ∠ BOC are complementary. This leads to the conclusion that the angles are not complementary.
∠ AOB and ∠ DOE are neither adjacent nor complementary angles.
Conclusion
Let's summarize the information we found from considering each pair of angles.
| Pair of Angles | Adjacent | Complementary |
|---|---|---|
| ∠ AOB and ∠ BOC | Yes | Yes |
| ∠ AOB and ∠ FOA | Yes | Yes |
| ∠ DOE and ∠ FOA | No | Yes |
| ∠ AOB and ∠ DOE | No | No |
Three pairs of angles share the characteristic of being complementary. This means that the pair of angles that are not complementary is the one that does not belong. For this reason, we can conclude that the pair ∠ AOB and ∠ DOE does not belong with the other three.
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Determine whether each of pair of lines is parallel, perpendicular, or neither.
{"type":"choice","form":{"alts":["Parallel","Perpendicular","Neither"],"noSort":true},"formTextBefore":"","formTextAfter":"","answer":2}
{"type":"choice","form":{"alts":["Parallel","Perpendicular","Neither"],"noSort":true},"formTextBefore":"","formTextAfter":"","answer":0}
{"type":"choice","form":{"alts":["Parallel","Perpendicular","Neither"],"noSort":true},"formTextBefore":"","formTextAfter":"","answer":1}
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By definition, parallel lines do not intersect each other. Since the given lines do intersect, we can state that they are not parallel. On the other hand, perpendicular lines intersect each other to form a right angle, which is denoted by a square angle marker. From the diagram, it looks like the lines may be perpendicular.
However, since there are no angle markers, we cannot determine for certain that the lines are perpendicular. For this reason, we conclude that the pair of lines are neither parallel nor perpendicular.
We can measure the angle between the lines using a protractor.
As we can see, the angle between the lines is 85^(∘), which is not a right angle. This is why there is no square angle marker in the diagram.
Notice that the given lines do not intersect each other, which suggests that the lines might be parallel. In addition, see that each line has a hatch mark.
The triangular hatch marks assure us that they are parallel lines.
Notice that these lines intersect each other, which means that they cannot be parallel. However, this time the angle formed by the lines is denoted by a square angle marker.
Therefore, the angle between the lines is a right angle. By definition, we can conclude that the lines are perpendicular.
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