Types of Angles

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 $60º$ is larger than one of $45º$, and two angles of $30º$ 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.

Angles are measured in degrees. A full circle represents $360°$ 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: $90°$.

In words: $90$ degrees.

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

It looks like this:

Acute angle, less than 90°

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

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

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

It looks like this:

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

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:

$∡CBA$ is smaller than: $∡DEF$

We will write it like this:

$∡CBA<∡DEF$

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).

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

They look 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.

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 equal.

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.

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

These two angles are adjacent angles.

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

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.

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.

Assignment

Among three parallel lines there are angles as sketched:

What is the value of $X$?

Solution

$AB\parallel CD\parallel EF$

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

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

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

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

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

Now we can calculate angle $3$

$180-99=81$

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

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

We solve the following equation to find $X$

$x=180-81-64$

$x=35$

Answer

$35^o$

Assignment

Is it possible to draw a quadrilateral that is not a rectangle in such a way that its opposite angles are equal?

Answer

True

Assignment

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

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

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

$CD>CE$

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

Solution

$EC$ and $BC$ are tangents to the circle

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

Now we calculate $AC$

$2R=AC$ diameter is the same

$A=16\pi=\pi r^2$

We take the square root

$R=\sqrt{16}=4$

$AC=2R=2\times4=8$

Now we calculate $CD$

$2R=CD$ diameter is the same

$P=8\pi=2\pi r$

We divide by: $2$

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

$CD=2R=2\times4=8$

From this we deduce that

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

Therefore $\triangle ABC\cong\triangle DEC$

By side, side, angle

Therefore $\sphericalangle BAC=\sphericalangle EDC$

Corresponding angles between congruent triangles are equal

Answer

Angle $CDE$ = Angle $BAC$

Assignment

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

Solution

$X=?$

$180^o-105^o=75^o$

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

$X=110^o -75^o$

$35^o$

Answer

$35^o$

Assignment

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

Solution

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

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

We move $32^o$ to the other side

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

Answer

$27^o$