Sum of Angles in a Polygon - Examples, Exercises and Solutions (2024)

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

Can these angles make a triangle?

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.

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.

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

Can these angles form a triangle?

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.

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

What kind of triangle is shown in the diagram below?

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.

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Triangle ABC isosceles.

AB = BC

Calculate angle ABC and indicate its type.

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:

$180-45-45=90$180−45−45=90

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

ABC is an isosceles triangle.

$∢A=4x$∢A=4x

$∢B=2x$∢B=2x

Calculate the value of x.

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:

$\frac{8X}{8}=\frac{180}{8}$88X​=8180​

$X=22.5$X=22.5

$∢C=95$∢C=95

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:

$B=360-220$B=360−220

$B=140$B=140

The angles below are between parallel lines.

What is the value of X?

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$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

$180-139=x$180−139=x

$x=41$x=41

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:

$360-310=B$360−310=B

$50=B$50=B

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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$360−90−90−90=90

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

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

$x=110-75=35$x=110−75=35

$\sphericalangle ABO\text{.}$∢ABO.

$∢\text{ABD}=15$∢ABD=15

BD bisects the angle.

Calculate the size of $∢\text{ABC}$∢ABC.

What is the value of the void angle?