Sum and difference of angles - Examples, Exercises and Solutions

In any polygon, you can calculate the sum of its internal angles using the following formula:

«Sum of the internal angles of a polygon» =180×(n2) =180\times\left(n-2\right)
while
n= n= «The number of edges or sides of the polygon» 

A1 - Sum of Angles in a Polygon

Steps to find the sum of the internal angles of a polygon:

  1. Count how many sides it has.
  2. Place it in the formula and we will obtain the sum of the internal angles of the polygon.

Important

In the formula, there are parentheses that require us to first perform the operations of subtraction (first we will subtract 2 2 from the number of edges and only then multiply by 180º 180º .

First of all, observe how many sides the given polygon has and write it as =n =n .
Then, note the correct n in the formula and discover the sum of the internal angles.

When it comes to a regular polygon (whose sides are all equal to each other) its angles will also be equal and we can calculate the size of each one of them.
For example, when it comes to a four-sided polygon (like a rectangle, rhombus, trapezoid, kite or diamond), the sum of its angles will be 360º 360º degrees.
However, when it comes to a polygon of 7 7 sides, the sum of its angles will be 900º 900º degrees. 

The sum of the external angles of a polygon will always be 360º 360º degrees.

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 Sum and difference of angles

Exercise #1

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 #2

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 #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

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:

18050=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 #5

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 #1

Find the measure of the angle α \alpha

505050AAABBBCCC50

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=180100 \alpha=180-100

α=80 \alpha=80

Answer

80

Exercise #2

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 #3

Triangle ABC isosceles.

AB = BC

Calculate angle ABC and indicate its type.

45°45°45°AAABBBCCC

Video Solution

Step-by-Step Solution

Given that it is an isosceles triangle:AB=BC AB=BC

It is possible to argue that:BAC=ACB=45 BAC=ACB=45

Since the sum of the angles of a triangle is 180, the angle ABC will be equal to:

1804545=90 180-45-45=90

Since the angle ABC measures 90 degrees, it is a right triangle.

Answer

90°, right angle.

Exercise #4

ABC is an isosceles triangle.

A=4x ∢A=4x

B=2x ∢B=2x

Calculate the value of x.

AAABBBCCC4x2x

Video Solution

Step-by-Step Solution

As we know that triangle ABC is isosceles.

B=C=2X B=C=2X

It is known that in a triangle the sum of the angles is 180.

Therefore, we can calculate in the following way:

2X+2X+4X=180 2X+2X+4X=180

4X+4X=180 4X+4X=180

8X=180 8X=180

We divide the two sections by 8:

8X8=1808 \frac{8X}{8}=\frac{180}{8}

X=22.5 X=22.5

Answer

22.5

Exercise #5

ABCD is a quadrilateral.

A=80 ∢A=80

C=95 ∢C=95

D=45 ∢D=45

Calculate the size of B ∢B .

AAABBBDDDCCC809545

Video Solution

Step-by-Step Solution

We know that the sum of the angles of a quadrilateral is 360°, that is:

A+B+C+D=360 A+B+C+D=360

We replace the known data within the following formula:

80+B+95+45=360 80+B+95+45=360

B+220=360 B+220=360

We move the integers to one side, making sure to keep the appropriate sign:

B=360220 B=360-220

B=140 B=140

Answer

140°

Exercise #1

It is known that angles A and D are equal to 90 degrees

Angle DEB is equal to 95 degrees

Complete the value of angle GDC based on the data from the figure.

505050404040707070AAABBBCCCDDDEEEFFFGGG3025

Video Solution

Step-by-Step Solution

Note that the GDC angle is part of the EDC angle.

Therefore, we can write the following expression:

GDC+EDG=EDC GDC+EDG=EDC

Since we know that angle D equals 90 degrees, we will substitute the values in the formula:

GDC+40=90 GDC+40=90

We will simplify the expression and keep the appropriate sign:

GDC=9040 GDC=90-40

GDC=50 GDC=50

Answer

50

Exercise #2

The angles below are between parallel lines.

XXX535353949494

What is the value of X?

Video Solution

Step-by-Step Solution

In the first step, we will have to find the adjacent angle of the 94 angle.

Let's remember that adjacent angles are equal to 180, therefore:

18094=86 180-94=86
Then let's observe the triangle.

Let's remember that the sum of the angles in a triangle is 180, therefore:

180=x+53+86 180=x+53+86

180=x+139 180=x+139

180139=x 180-139=x

x=41 x=41

Answer

41°

Exercise #3

ABCD is a quadrilateral.

According to the data, calculate the size of B ∢B .

AAABBBDDDCCC80140

Video Solution

Step-by-Step Solution

As we know, the sum of the angles in a square is equal to 360 degrees, therefore:

360=A+B+C+D 360=A+B+C+D

We replace the data we have in the previous formula:

360=140+B+80+90 360=140+B+80+90

360=310+B 360=310+B

Rearrange the sides and use the appropriate sign:

360310=B 360-310=B

50=B 50=B

Answer

50

Exercise #4

ABCD is a quadrilateral.

AB||CD
AC||BD

Calculate angle A ∢A .

90°90°90°AAABBBDDDCCC45°45°

Video Solution

Step-by-Step Solution

Angles ABC and DCB are alternate angles and equal to 45.

Angles ACB and DBC are alternate angles and equal to 45.

That is, angles B and C together equal 90 degrees.

Now we can calculate angle A, since we know that the sum of the angles of a square is 360:

360909090=90 360-90-90-90=90

Answer

90°

Exercise #5

110110110105105105XXX

What is the value of X given the angles between parallel lines shown above?

Video Solution

Step-by-Step Solution

Since the lines are parallel, we will draw another imaginary parallel line that crosses the angle of 110.

The angle adjacent to the angle 105 is equal to 75 (a straight angle is equal to 180 degrees) This angle is alternate with the angle that was divided using the imaginary line, therefore it is also equal to 75.

We are given that the whole angle is equal to 110 and we found only a part of it, we will indicate the second part of the angle as X since it changes and is equal to the existing angle X.

Now we can say that:

75+x=100 75+x=100

x=11075=35 x=110-75=35

Answer

35°

Topics learned in later sections

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