Types of Angles

🏆Practice sum and difference of angles

What is an angle?

Definition: Angles are created at the intersection between two lines. As seen in the following illustration

The angle in the illustration is called AB AB . We could also call it angle ABC \sphericalangle ABC . The important thing is that the middle letter is the one at the intersection of the lines.

For example, in this case:

The angle is BCD \sphericalangle BCD or DCB \sphericalangle DCB . Both notations are correct for the same angle.

We usually mark the angle with an arc as follows:

The marked angle is ABC ∡ABC . Sometimes we will denote angles using Greek letters, for example:

α α

or

β β

Before the name of the angle, we should note the angle symbol, like this:

Together it looks like this:

CBA ∡CBA

or

α ∡α

Next, we will delve into the size of angles, the different types, and those that are created when a line intersects two parallel lines.

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Given the equilateral triangle, find X

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There can be two angles that are equal, meaning they measure the same; likewise, a certain angle can be larger than another based on their measurements. 

For example, an angle of 60º 60º is larger than one of 45º 45º , and two angles of 30º 30º are equal. 

Angle larger than the other:

Angles of different sizes:

Notice that in these examples two angles were created, but at this stage, we will choose to refer to the acute angle (we will soon review what an acute angle is).

For example, in the following illustration:

Two angles were created as seen in the drawing:

At this phase, we will only refer to the acute angle of the two, the smaller one, the one that is between the two lines. This point might be a bit confusing, but don't worry because it will soon become clear to you.


How is an angle measured?

Angles are measured in degrees. A full circle represents 360° 360° degrees.

We will see this very clearly in the following illustration:

You can imagine that if we keep increasing the angle, we will eventually reach a full circle.

Whenever we want to indicate the size of an angle, we write the degree symbol next to the number. It is a small circle that is noted to the right of the number representing the angle size.

It looks like this: 90° 90° .

In words: 90 90 degrees.


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Acute Angle

Definition: An acute angle is one that measures less than 90° 90° :

It looks like this:


Right Angle

Definition: A right angle is one that measures exactly 90° 90° :

It looks like this:

Note that the marking of a right angle is not like that of other angles. It's not marked with an arc but with a symbol that looks like this


Do you know what the answer is?

Obtuse Angle

Definition: An obtuse angle is greater than 90° 90° and less than 180° 180° :

It looks like this:


Flat Angle

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Definition: A straight angle measures exactly 180° 180° .

It looks like this:

Next, we will learn how to calculate the size of angles. For now, we are satisfied with knowing that a right angle is larger than an acute angle, and that an obtuse angle is larger than a right angle. We understand this intuitively.

For example, this angle:

CBA ∡CBA is smaller than this one: DEF ∡DEF

We will write it like this:

CBA<DEF ∡CBA<∡DEF


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Opposite Angles by the Vertex

Definition: Vertically opposite angles are formed by two intersecting lines, with each pair facing each other.

For example:

The angles marked in red and also those in blue are vertically opposite. The angles in each pair of vertically opposite angles are equal (we'll delve deeper into this in other articles).


Angle Between Parallel Lines:

Definition: Recap: two parallel lines are lines that never meet.

It looks like this:

Line 1 and line 2 are parallel lines. Now we will draw another line, which crosses each of the parallel lines.

It looks like this:

That is, at the intersection between the two lines and the third, 8 angles were created (marked in the illustration). It is important to clarify that even if the lines were not parallel, 8 angles would be created. Now we will learn about the types of angles that have been created.


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Corresponding Angles

Definition: Corresponding angles are those that are on the same side of the transversal that cuts two parallel lines and are at the same level with respect to the parallel line. Corresponding angles are of identical size.

This definition might seem a bit confusing, but the illustration makes it very clear what corresponding angles are:

The two angles marked in red are corresponding angles. Therefore, they are also equal. Likewise, the angles marked in blue are also corresponding angles, meaning they are equal to each other. This is very important information that will help us later. Try to determine which angle is acute and which is obtuse.


Adjacent Angles

Definition: Adjacent angles are two angles that together form a straight angle (that is, 180° 180° ). Next, we will learn the meaning of the sum of angles.

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For example

These two angles are adjacent angles.

Another example

Notice, in this example the two angles marked in red are adjacent angles. Similarly, the angles marked in blue are also adjacent.


Do you know what the answer is?

Alternate Angles

Definition: Alternate angles are the ones that are on opposite sides of the transversal cutting through two parallel lines and are not on the same level with respect to the parallel line. Alternate angles are equal in size.

The explanation might be confusing, but the illustration makes it clear:

The two angles marked in blue are alternate angles, meaning they are also equal. The two angles marked in red are also alternate, and therefore, they are equivalent. Try to determine which angles are acute and which are obtuse.


Angle Types Exercises

Exercise 1

Assignment

Among three parallel lines there are angles as sketched:

What is the value of X X ?

Solution

ABCDEF AB\parallel CD\parallel EF

Let's focus on the line CD CD and extend its line to the left

We will mark the angle we create on that line with the number 1 1 and the existing angle which is equal to: 64o 64^o we will mark with the number 2 2

Now consider that angle 1 1 is equal to angle 2 2 since they are corresponding angles, therefore, angle 1 1 is also equal to: 64o 64^o

As the lines are parallel to each other, we will mark the angle next to the existing angle equal to: 99o 99^o with the number 3 3

Keep in mind that angle 3 3 and the angle 99o 99^o are adjacent angles, which means together they are equal to: 180o 180^o

Now we can calculate angle 3 3

18099=81 180-99=81

Now we have found 2 2 angles inside the triangle and we only need to calculate X X

As we know the sum of the angles in a triangle is 180o180^o

We solve the following equation to find X X

x=1808164 x=180-81-64

x=35 x=35

Answer

35o 35^o


Check your understanding

Exercise 2

Assignment

It is possible to draw a quadrilateral that is not a rectangle in such a way that its opposite angles are equal.

Solution

It is true that it is possible to draw a parallelogram that is a quadrilateral and not a rectangle, and in which the opposite angles are equal.

Answer

True


Exercise 3

Assignment

From point C C , two tangents are drawn to the circle O O

On AC AC , a semicircle is placed whose area is 16π 16\pi cm²

On CD CD , a semicircle is placed whose circumference is 8π 8\pi cm

CD>CE CD>CE

Which angles in the drawing are equal? (besides the given)

Solution

EC EC and BC BC are tangents to the circle

BC=EC BC=EC since tangents to a circle from the same point are equal

Now we calculate AC AC

2R=AC 2R=AC diameter is the same

A=16π=πr2 A=16\pi=\pi r^2

We take the square root

R=16=4 R=\sqrt{16}=4

AC=2R=2×4=8 AC=2R=2\times4=8

Now we calculate CD CD

2R=CD 2R=CD diameter is the same

P=8π=2πr P=8\pi=2\pi r

We divide by: 2 2

82=4=R \frac{8}{2}=4=R

CD=2R=2×4=8 CD=2R=2\times4=8

From this we deduce that

ABC=CED \sphericalangle ABC=\sphericalangle CED and BC=CE BC=CE , CD=AC CD=AC which is greater than CE CE

Therefore ABCDEC \triangle ABC\cong\triangle DEC

By side, side, angle

Therefore BAC=EDC \sphericalangle BAC=\sphericalangle EDC

Corresponding angles between congruent triangles are equal

Answer

Angle CDE CDE = Angle BAC BAC


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Exercise 4

Assignment

Given the angles between parallel lines in the graph, what is the value of: x x ?

Solution

X=? X=?

180o105o=75o 180^o-105^o=75^o

75o+X=110o 75^o+X=110^o /75o /-75^o

X=110o75o X=110^o -75^o

35o 35^o

Answer

35o 35^o


Exercise 5

Assignment

Given the angles between parallel lines as a sketch, what is the value of X X ?

Solution

There is a relationship of corresponding angles (corresponding angles) between the two angles, therefore they are equal.

Therefore, you can replace 59o 59^o as a result of the equation X+32=59 X+32=59

We move 32o 32^o to the other side

X=59o32o=27o X=59^o-32^o=27^o

Answer

27o 27^o


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