Types of Angles - Examples, Exercises and Solutions

Question Types:
Sum and Difference of Angles: Is it possible...?Sum and Difference of Angles: Isosceles triangleSum and Difference of Angles: Using variablesSum and Difference of Angles: Calculate Angles in QuadrilateralsSum and Difference of Angles: Identifying and defining elementsSum and Difference of Angles: Using quadrilateralsSum and Difference of Angles: Angle bisectorSum and Difference of Angles: Applying the formulaSum and Difference of Angles: Generate a random angleSum and Difference of Angles: Identify the greater valueSum and Difference of Angles: Using angles in a triangleSum and Difference of Angles: Finding the size of angles in a triangle

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.

Detailed explanation

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

Question 1

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

Can these angles form a triangle?

Question 2

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

Can these angles make a triangle?

Question 3

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

Can these angles form a triangle?

Question 4

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

XXXAAABBBCCC

Question 5

What is the value of the void angle?

160

Examples with solutions for Types of Angles

Exercise #1

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 #2

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 #3

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 #4

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 #5

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:

180−160=20 180-160=20

Answer

20

Question 1

Triangle ADE is similar to isosceles triangle ABC.

Angle A is equal to 50°.

Calculate angle D.

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Question 2

What is the size of angle ABC?

DBC = 100°

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Question 3

Can a triangle have more than one obtuse angle?

Question 4

What is the size of angle ABC?

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Question 5

Indicates which angle is greater

Exercise #6

Triangle ADE is similar to isosceles triangle ABC.

Angle A is equal to 50°.

Calculate angle D.

AAABBBCCCDDDEEE

Video Solution

Step-by-Step Solution

Triangle ABC is isosceles, therefore angle B is equal to angle C. We can calculate them since the sum of the angles of a triangle is 180:

180−50=130 180-50=130

130:2=65 130:2=65

As the triangles are similar, DE is parallel to BC

Angles B and D are corresponding and, therefore, are equal.

B=D=65

Answer

65 65 °

Exercise #7

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 #8

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 #9

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 #10

Indicates which angle is greater

Video Solution

Step-by-Step Solution

Note that in drawing A, the angle is more acute, meaning it's smaller:

While in drawing B, the angle is more obtuse, meaning it's larger:

Answer

Question 1

What kind of triangle is shown in the diagram below?

535353117117117212121AAABBBCCC

Question 2

Tree angles have the sizes 56°, 89°, and 17°.

Is it possible that these angles are in a triangle?

Question 3

Three angles measure as follows: 60°, 50°, and 70°.

Is it possible that these are angles in a triangle?

Question 4

Find the measure of the angle \( \alpha \)

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Question 5

Find the measure of the angle \( \alpha \)

120120120AAABBBCCC27

Exercise #11

What kind of triangle is shown in the diagram below?

535353117117117212121AAABBBCCC

Video Solution

Step-by-Step Solution

We calculate the sum of the angles of the triangle:

117+53+21=191 117+53+21=191

It seems that the sum of the angles of the triangle is not equal to 180°,

Therefore, the figure can not be a triangle and the drawing is incorrect.

Answer

The triangle is incorrect.

Exercise #12

Tree angles have the sizes 56°, 89°, and 17°.

Is it possible that these angles are in a triangle?

Video Solution

Step-by-Step Solution

Let's calculate the sum of the angles to see what total we get in this triangle:

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

The sum of angles in a triangle is 180 degrees, so this sum is not possible.

Answer

Impossible.

Exercise #13

Three angles measure as follows: 60°, 50°, and 70°.

Is it possible that these are angles in a triangle?

Video Solution

Step-by-Step Solution

Recall that the sum of angles in a triangle equals 180 degrees.

Let's add the three angles to see if their sum equals 180:

60+50+70=180 60+50+70=180

Therefore, it is possible that these are the values of angles in some triangle.

Answer

Possible.

Exercise #14

Find the measure of the angle α \alpha

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Video Solution

Step-by-Step Solution

Recall that the sum of angles in a triangle equals 180 degrees.

Therefore, we will use the following formula:

A+B+C=180 A+B+C=180

Now let's insert the known data:

α+50+50=180 \alpha+50+50=180

α+100=180 \alpha+100=180

We will simplify the expression and keep the appropriate sign:

α=180−100 \alpha=180-100

α=80 \alpha=80

Answer

80

Exercise #15

Find the measure of the angle α \alpha

120120120AAABBBCCC27

Video Solution

Step-by-Step Solution

Let's remember that the sum of angles in a triangle is equal to 180 degrees.

Therefore, we will use the following formula:

A+B+C=180 A+B+C=180

Now let's input the known data:

120+27+α=180 120+27+\alpha=180

147+α=180 147+\alpha=180

We'll move the term to the other side and keep the appropriate sign:

α=180−147 \alpha=180-147

α=33 \alpha=33

Answer

33

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