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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

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.
Applet where a spiral notebook can be opened. The angle between the bottom part and the grabbed part is shown along with its measure.
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.

Angle ABC is formed by two rays, BC and BA, which are referred to as the sides of the angle. The starting points of both rays are identical, originating from point B, which serves as the vertex 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
or or or

The measure of an angle, denoted by 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
These regions can be examined in the following graph.
Interior and Exterior of an angle
Notice that the interior of the angle is the region for which the angle measure is less than
Discussion

Classifying an Angle From Its Measure

The measure of an angle can range from to or from to 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 but less than

Applet showing when an angle is acute and when it is not.
Discussion

Angles With a Measure of

Going from least to great, the second type of angle occurs when the measure is exactly

Concept

Right Angle

A right angle is an angle whose measure is exactly In diagrams, right angles are denoted with a square angle marker.

Applet showing when an angle is acute and when it is not.
Discussion

Angles With a Measure Ranging From to

The third type of angle groups all those angles whose measure is greater than but less than

Concept

Obtuse Angle

An obtuse angle is an angle whose measure is greater than but less than

Applet showing when an angle is acute and when it is not.
Discussion

Angles With a Measure of

As with right angles, the following type of angle involves only those angles whose measure is exactly

Concept

Straight Angle

A straight angle is an angle whose measure is exactly

Applet showing when an angle is acute and when it is not.
Discussion

Angles With a Measure Greater Than

The fifth type of angle includes angles whose measure is greater than but less than This is the largest range of measures.

Concept

Reflexive Angle

A reflexive angle is an angle whose measure is greater than but less than An alternative name for this type of angle is reflex angle.

Applet showing when an angle is acute and when it is not.
Discussion

Angles With a Measure of

The last type of angle is formed by all the angles whose measure is exactly

Concept

Complete Angle

A complete angle is an angle whose measure is exactly Alternative names for this type of angle are full angle, round angle, and perigon.

Applet showing when an angle is acute and when it is not.
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.

A clock and the angle between the minute hand and the hour hand is labeled.
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.

A laser beam placed diagonally above a horizontally positioned mirror creating an incident beam and a reflected beam at the same angle with respect to the mirror.

In the diagram, and have the same measure. Angles with the same measure have a special name.

Concept

Congruent Angles

Two angles are congruent angles if both have the same measure. In a diagram, congruent angles are usually indicated by the same number of angle markers.
Two congruent angles
The symbol is used to algebraically express that two angles are congruent.
Discussion

Classifying Angle Pairs Based on Their Position

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

Concept

Adjacent Angles

Two angles are adjacent angles if they meet the following conditions.

  • They share a common side.
  • They share a common vertex.
  • They do not overlap.

The diagram illustrates an example of adjacent angles.

Two adjacent acute angles, angle BOA and angle COB. The measure of angle BOA is represented by alpha and the measure of angle COB is represented by beta.
Angle and angle share the vertex and the side with no overlapping region.

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.

A beam reflected in a mirror
The incident beam is a common side to and so these angles are adjacent. Similarly, and 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

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.

Two intersecting line segments and four angles.

Vertical angles are always congruent.

Two intersecting lines that form two pairs of vertical angles

Based on the characteristics of the diagram, the following relations hold true.


Example

Naming Pairs of Angles

In the diagram, lines and intersect at point and is a point on the interior of

Diagram of lines AD and CE intersecting at X. Ray XB divides angle AXC into two angles with the same measure.
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 and
c What is the measure of

Answer
a
Congruent Angles Adjacent Angles Vertical Angles
and
and
and 		and
and
and
and
and 		and
and
b There is no relation between and
c

Hint
a Congruent angles are denoted with the same number of angle markers. Adjacent angles have the same vertex, share one side, and they do not overlap. Vertical angles are formed when two lines intersect.
b Remove the unnecessary parts of the diagram to have a better look.
c Use the fact that measures

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.
Diagram of lines AD and CE intersecting at X. Ray XB divides angle AXC into two angles with the same measure. Angles with the same number of hatch marks are pointed out

Since and have the same number of markers, the angles are congruent.

Congruent Angles
and

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 and

Diagram of lines AD and CE intersecting at X. Ray XB divides angle AXC into two angles with the same measure. The vertex X and the common side XE are pointed out.

In the diagram there are five pairs of adjacent angles.

Adjacent Angles
and
and
and
and
and

Finally, look for vertical angles. Vertical angles are opposite angles formed when two lines or line segments intersect. Since lines and intersect at 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.

Diagram of lines AD and CE intersecting at X. Ray XB divides angle AXC into two angles with the same measure.

From the diagram, and are vertical angles, as are and

Vertical Angles
and
and

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
and
and
and 		and
and
and
and
and 		and
and
b To check whether and are related, ignore the irrelevant parts for a moment.
Diagram of lines AD and CE intersecting at X. Ray XB divides angle AXC into two angles with the same measure.

The angles have the same vertex but they do not have a common side. Therefore, they are not adjacent angles. Notice that has no angle marker and its measure seems to be greater than the measure of Therefore, the angles are not congruent.

and
Adjacent Congruent Vertical
${\color{#FF0000}{\bm{\Large{\times }}}}$ ${\color{#FF0000}{\bm{\Large{\times }}}}$

Lastly, note that and lie on the same line but and do not. Therefore, and are not vertical angles. As such, there is no relation between these angles.

and
Adjacent Congruent Vertical
${\color{#FF0000}{\bm{\Large{\times }}}}$ ${\color{#FF0000}{\bm{\Large{\times }}}}$ ${\color{#FF0000}{\bm{\Large{\times }}}}$
c From Part A, and are vertical angles, which also means they are congruent. This means that they have the same measure. Since the measure of is the measure of is also
Diagram of lines AD and CE intersecting at X. Ray XB divides angle AXC into two angles with the same measure.
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
Applet showing two angles for which the sum of the measures is 90
In the diagram, and are complementary because the sum of their measures is equal to

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.

A pair of complementary angles that are put together so they form a right angle
Example

Angle Formed by the Clock Hands

When the clock shows and seconds, the angle between the minute hand and the second hand is while the angle between the minute hand and the hour hand is

A clock. The hour hand points to 9. The minute's hand points to 12. The second's hand points to 11.
If the angle between the second hand and the hour hand is what is the value of

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.

A clock. The hour hand points to 9. The minutes hand points to 12. The seconds hand points to 11.
Note that and together make a right angle. As a result, these angles are complementary. This means that the sum of their measures is
From the diagram, and Substitute these values into the equation and solve for
SubstituteII

,

▼
Solve for
SubEqn

SubTerm

Subtract term

DivEqn

CalcQuot

Calculate quotient

Discussion

Supplementary Angles

Two angles are supplementary angles when the sum of their measures is
Applet showing two angles for which the sum of the measures is 90
In the diagram, and are supplementary because the sum of their measures equals

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.

A pair of supplementary angles that are put together so they form a straight angle
Example

Computer Screen Tilt

Some studies recommend tilting the computer screen 💻 slightly backwards between and for better posture and range of vision.

A laptop over a table. The screen makes two angles with the table; one measures 110 degrees and the other 2x+30 degrees.
In the diagram, what is the value of

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.

A laptop over a table. The screen makes two angles with the table; one measures 110 degrees and the other 2x+30 degrees.
For that reason, the measures of these angles add up to
From the diagram, and Substitute these measures into the equation and solve for
SubstituteII

,

▼
Solve for
AddTerms

Add terms

SubEqn

SubTerm

Subtract term

DivEqn

CalcQuot

Calculate quotient

Discussion

Explementary Angles

Two angles are explementary angles, also called conjugate angles, when the sum of their measures is
Applet showing two angles for which the sum of the measures is 90
In the diagram, and are explementary because the sum of their measures is

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.

A pair of explementary angles that are put together so they form a straight 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

Binocular vision of a human seen from the top.
Calculate the value of

Hint

The binocular vision angle and the blind spot angle are explementary angles.

Solution

For simplicity, mark some points in the given diagram.

Binocular vision of a human seen from the top.
Notice that and together make a complete angle. Consequently, they are explementary angles. As such, the sum of their measures is
From the diagram, the binocular vision angle measures and the blind spot angle measures Substitute for and for and solve the equation for
SubstituteII

,

▼
Solve for
AddTerms

Add terms

SubEqn

SubTerm

Subtract term

DivEqn

CalcQuot

Calculate quotient

If it can be also found that the blind spot angle measures
Pop Quiz

Classifying Pairs of Angles

Classify each given pair of angles as complementary, supplementary, or explementary angles, or if they have no relationship.

Random pairs of angles with the measures labeled
Pop Quiz

Finding Angle Measures

For each of the given diagrams, find the value of

A random pair of angles (vertical, complementary, supplementary, explementary, congruent angles). One measure is given and the other is required.
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 and are parallel.

One pair of parallel lines and one transversal
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 and are perpendicular.

Two lines intersecting at a right angle
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.

Two random lines
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.
Two lines cut by a third line
At each intersection point there are two pairs of vertical angles and four pairs of supplementary angles.
Vertical Angles Supplementary Angles
and
and 		and and
and and
and
and 		and and
and and
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.
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