What is an angle?

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

Angles are created at the intersection between two lines

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:

We could also call it angle ABC

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 angle is BCD

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.

Suggested Topics to Practice in Advance

  1. Right angle
  2. Acute Angles
  3. Obtuse Angle
  4. Plane angle
  5. Angle Notation
  6. Angle Bisector

Practice Types of Angles

Examples with solutions for Types of Angles

Exercise #1

In a right triangle, the sum of the two non-right angles is...?

Video Solution

Step-by-Step Solution

In a right-angled triangle, there is one angle that equals 90 degrees, and the other two angles sum up to 180 degrees (sum of angles in a triangle)

Therefore, the sum of the two non-right angles is 90 degrees

90+90=180 90+90=180

Answer

90 degrees

Exercise #2

Calculate the size of angle X given that the triangle is equilateral.

XXXAAABBBCCC

Video Solution

Step-by-Step Solution

Remember that the sum of angles in a triangle is equal to 180.

In an equilateral triangle, all sides and all angles are equal to each other.

Therefore, we will calculate as follows:

x+x+x=180 x+x+x=180

3x=180 3x=180

We divide both sides by 3:

x=60 x=60

Answer

60

Exercise #3

What is the value of the void angle?

160

Video Solution

Step-by-Step Solution

The empty angle is an angle adjacent to 160 degrees.

Remember that the sum of adjacent angles is 180 degrees.

Therefore, the value of the empty angle will be:

180160=20 180-160=20

Answer

20

Exercise #4

What type of angle is α \alpha ?

αα

Step-by-Step Solution

Let's remember that an acute angle is smaller than 90 degrees, an obtuse angle is larger than 90 degrees, and a straight angle equals 180 degrees.

Since in the drawing we have lines perpendicular to each other, the marked angles are right angles, each equal to 90 degrees.

Answer

Straight

Exercise #5

Angle A is equal to 30°.
Angle B is equal to 60°.
Angle C is equal to 90°.

Can these angles form a triangle?

Video Solution

Step-by-Step Solution

We add the three angles to see if they equal 180 degrees:

30+60+90=180 30+60+90=180
The sum of the angles equals 180, so they can form a triangle.

Answer

Yes

Exercise #6

Angle A equals 90°.
Angle B equals 115°.
Angle C equals 35°.

Can these angles form a triangle?

Video Solution

Step-by-Step Solution

We add the three angles to see if they are equal to 180 degrees:

90+115+35=240 90+115+35=240
The sum of the given angles is not equal to 180, so they cannot form a triangle.

Answer

No.

Exercise #7

Angle A equals 56°.
Angle B equals 89°.
Angle C equals 17°.

Can these angles make a triangle?

Video Solution

Step-by-Step Solution

We add the three angles to see if they are equal to 180 degrees:

56+89+17=162 56+89+17=162

The sum of the given angles is not equal to 180, so they cannot form a triangle.

Answer

No.

Exercise #8

Below is an equilateral triangle.

Calculate X.

X+5X+5X+5AAABBBCCC

Video Solution

Step-by-Step Solution

Since in an equilateral triangle all sides are equal and all angles are equal. It is also known that in a triangle the sum of angles is 180°, we can calculate X in the following way:

X+5+X+5+X+5=180 X+5+X+5+X+5=180

3X+15=180 3X+15=180

3X=18015 3X=180-15

3X=165 3X=165

Let's divide both sides by 3:

3X3=1653 \frac{3X}{3}=\frac{165}{3}

X=55 X=55

Answer

55

Exercise #9

What is the size of angle ABC?

DBC = 100°

DDDBBBCCCAAA10040

Video Solution

Step-by-Step Solution

Given that angle DBC is equal to 100 degrees. Let's look at the letters and note that angle ABC is given and equals 40 degrees

Answer

40

Exercise #10

Indicates which angle is greater

Video Solution

Step-by-Step Solution

Answer B is correct because the more closed the angle is, the more acute it is (less than 90 degrees), meaning it's smaller.

The more open the angle is, the more obtuse it is (greater than 90 degrees), meaning it's larger.

Answer

Exercise #11

ABC is an equilateral triangle.8X8X8XAAABBBCCCCalculate X.

Video Solution

Step-by-Step Solution

Since this is an equilateral triangle, all angles are also equal.

As the sum of angles in a triangle is 180 degrees, each angle is equal to 60 degrees. (180:3=60)

From this, we can conclude that: 60=8x 60=8x

Let's divide both sides by 8:

608=8x8 \frac{60}{8}=\frac{8x}{8}

7.5=x 7.5=x

Answer

7.5

Exercise #12

Can a triangle have more than one obtuse angle?

Video Solution

Step-by-Step Solution

If we try to draw two obtuse angles and connect them to form a triangle (i.e., only 3 sides), we will see that it is not possible.

Therefore, the answer is no.

Answer

No

Exercise #13

What is the size of angle ABC?

AAABBBCCC6050

Video Solution

Step-by-Step Solution

In order to calculate the value of angle ABC, we must calculate the sum of all the given angles.

That is:

ABC=60+50 ABC=60+50

ABC=110 ABC=110

Answer

110

Exercise #14

Indicates which angle is greater

Video Solution

Step-by-Step Solution

In drawing A, we can see that the angle is an obtuse angle, meaning it is larger than 90 degrees:

While in drawing B, the angle is a right angle, meaning it equals 90 degrees:

Therefore, the larger angle appears in drawing A.

Answer

Exercise #15

Indicates which angle is greater

Video Solution

Step-by-Step Solution

In drawing A, we can see that the angle is more closed:

While in drawing B, the angle is more open:

In other words, in drawing A the angle is more acute, while in drawing B the angle is more obtuse.

We'll remember that the more obtuse an angle is, the larger it is.

Therefore, the larger angle appears in drawing B.

Answer

Topics learned in later sections

  1. Sides, Vertices, and Angles
  2. Sum and Difference of Angles
  3. Sum of Angles in a Polygon
  4. The Sum of the Interior Angles of a Triangle
  5. Exterior angles of a triangle