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»$=180\times\left(n-2\right)$ while$n=$ «The number of edges or sides of the 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 $2$ 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.

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?

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.

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?

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.

Yes

Exercise #3

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

Can these angles form a triangle?

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.

No.

Exercise #4

Triangle ADE is similar to isosceles triangle ABC.

Angle A is equal to 50°.

Calculate angle D.

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

$65$°

Exercise #5

What kind of triangle is shown in the diagram below?

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.

The triangle is incorrect.

Exercise #1

Find the measure of the angle $\alpha$

Step-by-Step Solution

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

Therefore, we will use the following formula:

$A+B+C=180$

Now let's insert the known data:

$\alpha+50+50=180$

$\alpha+100=180$

We will simplify the expression and keep the appropriate sign:

$\alpha=180-100$

$\alpha=80$

80

Exercise #2

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

Is it possible that these are angles in a triangle?

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:

$60+50+70=180$

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

Possible.

Exercise #3

Triangle ABC isosceles.

AB = BC

Calculate angle ABC and indicate its type.

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.

90°, right angle.

Exercise #4

ABC is an isosceles triangle.

$∢A=4x$

$∢B=2x$

Calculate the value of x.

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:

$\frac{8X}{8}=\frac{180}{8}$

$X=22.5$

22.5

Exercise #5

$∢A=80$

$∢C=95$

$∢D=45$

Calculate the size of $∢B$.

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:

$B=360-220$

$B=140$

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.

Step-by-Step Solution

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

Therefore, we can write the following expression:

$GDC+EDG=EDC$

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

$GDC+40=90$

We will simplify the expression and keep the appropriate sign:

$GDC=90-40$

$GDC=50$

50

Exercise #2

The angles below are between parallel lines.

What is the value of X?

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:

$180=x+53+86$

$180=x+139$

$180-139=x$

$x=41$

41°

Exercise #3

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

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:

$360-310=B$

$50=B$

50

Exercise #4

AB||CD
AC||BD

Calculate angle $∢A$.

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:

$360-90-90-90=90$

90°

Exercise #5

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

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:

$75+x=100$

$x=110-75=35$

35°