Examples with solutions for Sum and Difference of Angles: Finding the size of angles in a triangle

Exercise #1

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 #2

Below is an equilateral triangle.

Calculate X.

X+5X+5X+5AAABBBCCC

Video Solution

Step-by-Step Solution

Since in an equilateral triangle all sides are equal and all angles are equal. It is also known that in a triangle the sum of angles is 180°, we can calculate X in the following way:

X+5+X+5+X+5=180 X+5+X+5+X+5=180

3X+15=180 3X+15=180

3X=18015 3X=180-15

3X=165 3X=165

Let's divide both sides by 3:

3X3=1653 \frac{3X}{3}=\frac{165}{3}

X=55 X=55

Answer

55

Exercise #3

ABC is an equilateral triangle.8X8X8XAAABBBCCCCalculate X.

Video Solution

Step-by-Step Solution

Since this is an equilateral triangle, all angles are also equal.

As the sum of angles in a triangle is 180 degrees, each angle is equal to 60 degrees. (180:3=60)

From this, we can conclude that: 60=8x 60=8x

Let's divide both sides by 8:

608=8x8 \frac{60}{8}=\frac{8x}{8}

7.5=x 7.5=x

Answer

7.5

Exercise #4

Find the measure of the angle α \alpha

949494AAABBBCCC92

Video Solution

Step-by-Step Solution

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

Since we are given two angles, we can calculate a a

94+92=186 94+92=186

We should note that the sum of the two given angles is greater than 180 degrees.

Therefore, there is no solution possible.

Answer

There is no possibility of resolving

Exercise #5

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 #6

What type of triangle appears in the drawing?

ααα303030101010

Video Solution

Step-by-Step Solution

To determine which type of triangle we are dealing with, let's calculate angle alpha based on the fact that the sum of angles in a triangle is 180 degrees.

α=1803010=140 \alpha=180-30-10=140

Since alpha is equal to 140 degrees, the triangle is an obtuse triangle.

Answer

Obtuse triangle

Exercise #7

Find all the angles of the isosceles triangle using the data in the figure.

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Video Solution

Step-by-Step Solution

In an isosceles triangle, the base angles are equal to each other, meaning:

B=C B=C

Since we are given angle A, we can calculate the base angles as follows:

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

18050=130 180-50=130

130:2=65 130:2=65

B=C=65 B=C=65

Answer

B=65,C=65 B=65,C=65

Exercise #8

Find all the angles of the isosceles triangle using the data in the figure.

505050AAACCCBBB

Video Solution

Step-by-Step Solution

Since we are given that the triangle is isosceles, we will remember that the base angles are equal to each other.

That is:

B=C=50 B=C=50

Now we can calculate the vertex angle.

Since the sum of angles in a triangle is equal to 180 degrees, we will calculate the vertex angle as follows:

A=1805050=80 A=180-50-50=80

Therefore, the values of the angles in the triangle are: 80, 50, 50

Answer

A=80,C=50 A=80,C=50

Exercise #9

Find all the angles of the isosceles triangle using the data in the figure.

707070AAABBBCCC

Video Solution

Step-by-Step Solution

Let's remember that in an isosceles triangle, the base angles are equal to each other.

In other words:

C=B C=B

Since we are given the vertex angle, which is equal to 70 degrees, we'll recall that the sum of angles in a triangle is equal to 180 degrees.

Now let's find the base angles in the following way:

18070=110 180-70=110

110:2=55 110:2=55

Therefore, the angle values in the triangle are: 55, 55, 70

Answer

70, 55, 55

Exercise #10

Find all the angles of the isosceles triangle using the data in the figure.

626262AAABBBCCC

Video Solution

Step-by-Step Solution

In an isosceles triangle, we remember that the base angles are equal to each other, so angles C and B are equal to each other:

C=B=62 C=B=62

Now we can calculate the vertex angle.

We remember that the sum of angles in a triangle is equal to 180 degrees, therefore:

A=1806262=56 A=180-62-62=56

The angle values in the triangle are: 62, 62, 56

Answer

62, 62, 56

Exercise #11

Find all the angles of the isosceles triangle using the data in the figure.

Video Solution

Step-by-Step Solution

In the triangle shown in the diagram, we notice that one angle is a right angle equal to 90 degrees.

We'll remember that in an isosceles right triangle, the base angles are equal to each other.

Since the sum of angles in a triangle is equal to 180, we can calculate the angles as follows:

18090=90 180-90=90

90:2=45 90:2=45

Therefore, the angle values in the triangle are: 90, 45, 45

Answer

90, 45, 45

Exercise #12

Look at the isosceles right triangle below. What are its angles?

Video Solution

Step-by-Step Solution

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

In a right triangle, there is one right angle equal to 90 degrees.

In an isosceles triangle, the base angles are equal to each other.

Therefore, we can calculate this in the following way:

18090=90 180-90=90

90:2=45 90:2=45

In other words, the angle values in this triangle are: 90, 45, 45

Answer

90, 45, 45

Exercise #13

Tree angles have the sizes:

90°, 60°, and 30.

Is it possible that these angles are in a triangle?

Video Solution

Step-by-Step Solution

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

We'll add the three angles to see if their sum equals 180:

90+60+30=180 90+60+30=180

Therefore, these could be the values of angles in some triangle.

Answer

No.

Exercise #14

Tree angles have the sizes:

50°, 41°, and 81.

Is it possible that these angles are in a triangle?


Video Solution

Step-by-Step Solution

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

We'll add the three angles to see if their sum equals 180:

50+41+81=172 50+41+81=172

Therefore, these cannot be the values of angles in any triangle.

Answer

Impossible.

Exercise #15

Tree angles have the sizes:

69°, 93°, and 81.

Is it possible that these angles are in a triangle?

Video Solution

Step-by-Step Solution

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

We'll add the three angles to see if their sum equals 180:

69+81+93=243 69+81+93=243

Therefore, these cannot be the values of angles in any triangle.

Answer

No.

Exercise #16

Tree angles have the sizes:

76°, 52°, and 52°.

Is it possible that these angles are in a triangle?

Video Solution

Step-by-Step Solution

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

We will add the three angles to find out if their sum equals 180:

76+52+52=180 76+52+52=180

Therefore, these could be the values of angles in some triangle.

Answer

Yes.

Exercise #17

Tree angles have the sizes:

31°, 122°, and 85.

Is it possible that these angles are in a triangle?

Video Solution

Step-by-Step Solution

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

We'll add the three angles to see if their sum equals 180:

31+122+85=238 31+122+85=238

Therefore, these cannot be the values of angles in any triangle.

Answer

Impossible.

Exercise #18

Tree angles have the sizes 94°, 36.5°, and 49.5. Is it possible that these angles are in a triangle?

Video Solution

Step-by-Step Solution

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

We'll add the three angles to see if their sum equals 180:

94+36.5+49.5=180 94+36.5+49.5=180

Therefore, these could be the values of angles in some triangle.

Answer

Possible.

Exercise #19

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 #20

Tree angles have the sizes:

90°, 60°, and 40.

Is it possible that these angles are in a triangle?

Video Solution

Step-by-Step Solution

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

We'll add the three angles to see if their sum equals 180:

90+60+40=190 90+60+40=190

Therefore, these cannot be the values of angles in any triangle.

Answer

Yes.