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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

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

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.

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

Notice that the interior of the angle is the region for which the angle measure is less than $180_{∘}.$

Discussion

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

An acute angle is an angle whose measure is greater than $0_{∘}$ but less than $90_{∘}.$

Discussion

Discussion

Discussion

As with right angles, the following type of angle involves only those angles whose measure is exactly $180_{∘}.$

Concept

A straight angle is an angle whose measure is exactly $180_{∘}.$

Discussion

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

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

Pop Quiz

As time passes, the hands of a clock form different angles. Classify the indicated angle by estimating its measure.

Discussion

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

Discussion

Angles can also be classified based on their position relative to other angles.

Concept

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.

The incident beam is a common side to $∠1$ and $∠2,$ so these angles are adjacent. Similarly, $∠2$ and $∠3$ are also adjacent angles because they share the side made by the reflected beam. Next, a third way of relating a pair of angles is presented.Discussion

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

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 $∠EXA?$

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 $m∠EXA=100_{∘}$

b Remove the unnecessary parts of the diagram to have a better look.

c Use the fact that $∠CXD$ measures $100_{∘}.$

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

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

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

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 $90_{∘}.$

If the angle between the second hand and the hour hand is $5x_{∘},$ what is the value of $x?${"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":[],"constants":[]}},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.43056em;vertical-align:0em;\"><\/span><span class=\"mord mathdefault\">x<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">=<\/span><\/span><\/span><\/span>","formTextAfter":null,"answer":{"text":["12"]}}

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.

Start by marking a few points on the diagram to make it easy to reference its parts.

Note that $∠JKL$ and $∠LKM$ together make a right angle. As a result, these angles are complementary. This means that the sum of their measures is $90_{∘}.$$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

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

Some studies recommend tilting the computer screen 💻 slightly backwards between $10_{∘}$ and $20_{∘}$ for better posture and range of vision.

In the diagram, what is the value of $x?${"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":[],"constants":[]}},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.43056em;vertical-align:0em;\"><\/span><span class=\"mord mathdefault\">x<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">=<\/span><\/span><\/span><\/span>","formTextAfter":null,"answer":{"text":["20"]}}

The computer screen forms two supplementary angles with the table.

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.

For that reason, the measures of these angles add up to $180_{∘}.$$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