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×(n−2)
while
n= «The number of edges or sides of the polygon»
![Sum and difference of angles - Examples, Exercises and Solutions (1) Sum and difference of angles - Examples, Exercises and Solutions (1)](https://i0.wp.com/www.tutorela.com/_ipx/f_png,s_500x499/cdn.tutorela.com/images/A1_-_Sum_of_Angles_in_a_Polygon.width-500.png)
Steps to find the sum of the internal angles of a polygon:
- Count how many sides it has.
- 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 from the number of edges and only then multiply by 180º.
First of all, observe how many sides the given polygon has and write it as =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 degrees.
However, when it comes to a polygon of 7 sides, the sum of its angles will be 900º degrees.
The sum of the external angles of a polygon will always be 360º degrees.
Detailed explanation
Suggested Topics to Practice in Advance
- Right angle
- Acute Angles
- Obtuse Angle
- Plane angle
- Angle Notation
- Angle Bisector
Practice Sum and difference of angles
Question 1
Angle A equals 56°.
Angle B equals 89°.
Angle C equals 17°.
Can these angles make a triangle?
Question 2
Angle A equals 90°.
Angle B equals 115°.
Angle C equals 35°.
Can these angles form a triangle?
Question 3
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 4
Triangle ADE is similar to isosceles triangle ABC.
Angle A is equal to 50°.
Calculate angle D.
Question 5
What kind of triangle is shown in the diagram below?
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
The sum of the given angles is not equal to 180, so they cannot form a triangle.
Answer
No.
Exercise #2
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
The sum of the given angles is not equal to 180, so they cannot form a triangle.
Answer
No.
Exercise #3
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
The sum of the angles equals 180, so they can form a triangle.
Answer
Yes
Exercise #4
Triangle ADE is similar to isosceles triangle ABC.
Angle A is equal to 50°.
Calculate angle D.
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
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°
Exercise #5
What kind of triangle is shown in the diagram below?
Video Solution
Step-by-Step Solution
We calculate the sum of the angles of the triangle:
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.
Question 1
ABC is an isosceles triangle.
\( ∢A=4x \)
\( ∢B=2x \)
Calculate the value of x.
Question 2
Triangle ABC isosceles.
AB = BC
Calculate angle ABC and indicate its type.
Question 3
ABCD is a quadrilateral.
\( ∢A=80 \)
\( ∢C=95 \)
\( ∢D=45 \)
Calculate the size of \( ∢B \).
Question 4
ABCD is a quadrilateral.
According to the data, calculate the size of \( ∢B \).
Question 5
The angles below are between parallel lines.
What is the value of X?
Exercise #1
ABC is an isosceles triangle.
∢A=4x
∢B=2x
Calculate the value of x.
Video Solution
Step-by-Step Solution
As we know that triangle ABC is isosceles.
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
4X+4X=180
8X=180
We divide the two sections by 8:
88X=8180
X=22.5
Answer
22.5
Exercise #2
Triangle ABC isosceles.
AB = BC
Calculate angle ABC and indicate its type.
Video Solution
Step-by-Step Solution
Given that it is an isosceles triangle:AB=BC
It is possible to argue that:BAC=ACB=45
Since the sum of the angles of a triangle is 180, the angle ABC will be equal to:
180−45−45=90
Since the angle ABC measures 90 degrees, it is a right triangle.
Answer
90°, right angle.
Exercise #3
ABCD is a quadrilateral.
∢A=80
∢C=95
∢D=45
Calculate the size of ∢B.
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
We replace the known data within the following formula:
80+B+95+45=360
B+220=360
We move the integers to one side, making sure to keep the appropriate sign:
B=360−220
B=140
Answer
140°
Exercise #4
ABCD is a quadrilateral.
According to the data, calculate the size of ∢B.
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
We replace the data we have in the previous formula:
360=140+B+80+90
360=310+B
Rearrange the sides and use the appropriate sign:
360−310=B
50=B
Answer
50
Exercise #5
The angles below are between parallel lines.
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:
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+139
180−139=x
x=41
Answer
41°
Question 1
ABCD is a quadrilateral.
AB||CD
AC||BD
Calculate angle \( ∢A \).
Question 2
What is the value of X given the angles between parallel lines shown above?
Question 3
In a right triangle, the sum of the two non-right angles is...?
Question 4
What is the value of the void angle?
Question 5
Calculate the size of angle X given that the triangle is equilateral.
Exercise #1
ABCD is a quadrilateral.
AB||CD
AC||BD
Calculate angle ∢A.
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:
360−90−90−90=90
Answer
90°
Exercise #2
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
x=110−75=35
Answer
35°
Exercise #3
In a right triangle, the sum of the two non-right angles is...?
Video Solution
Answer
90 degrees
Exercise #4
What is the value of the void angle?
Video Solution
Answer
20
Exercise #5
Calculate the size of angle X given that the triangle is equilateral.
Video Solution
Answer
60
Topics learned in later sections
- The Sum of the Interior Angles of a Triangle
- Sides, Vertices, and Angles
- Types of Angles
- Sum and Difference of Angles
- Exterior angle of a triangle