Examples with solutions for Parts of a Triangle: Finding the size of angles in a triangle

Exercise #1

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 #2

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 #3

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 #4

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 #5

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 #6

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 #7

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 #8

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 #9

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 #10

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.

Exercise #11

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 #12

Find the measure of the angle α \alpha

120120120AAABBBCCC27

Video Solution

Step-by-Step Solution

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

Therefore, we will use the following formula:

A+B+C=180 A+B+C=180

Now let's input the known data:

120+27+α=180 120+27+\alpha=180

147+α=180 147+\alpha=180

We'll move the term to the other side and keep the appropriate sign:

α=180147 \alpha=180-147

α=33 \alpha=33

Answer

33

Exercise #13

Find the measure of the angle α \alpha

808080AAABBBCCC55

Video Solution

Step-by-Step Solution

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

Therefore, we will use the following formula:

A+B+C=180 A+B+C=180

Now let's input the known data:

80+55+α=180 80+55+\alpha=180

135+α=180 135+\alpha=180

We'll move the term to the other side and keep the appropriate sign:

α=180135 \alpha=180-135

α=45 \alpha=45

Answer

45

Exercise #14

Find the measure of the angle α \alpha

696969AAABBBCCC23

Video Solution

Step-by-Step Solution

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

Therefore, we will use the following formula:

A+B+C=180 A+B+C=180

Now let's input the known data:

α+69+23=180 \alpha+69+23=180

α+92=180 \alpha+92=180

We'll move the term to the other side and keep the appropriate sign:

α=18092 \alpha=180-92

α=88 \alpha=88

Answer

88

Exercise #15

Find the measure of the angle α \alpha

27.727.727.7AAABBBCCC41

Video Solution

Step-by-Step Solution

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

Therefore, we will use the formula:

A+B+C=180 A+B+C=180

We will substitute the known data:

α+27.7+41=180 \alpha+27.7+41=180

α+68.7=180 \alpha+68.7=180

We will move the term to the other side and maintain the appropriate sign:

α=18068.7 \alpha=180-68.7

α=111.3 \alpha=111.3

Answer

111.3

Exercise #16

Find the measure of the angle α \alpha

100100100AAABBBCCC90

Video Solution

Step-by-Step Solution

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

Therefore, we will use the formula:

A+B+C=180 A+B+C=180

Let's input the known data:

100+α+90=180 100+\alpha+90=180

190+α=180 190+\alpha=180

α=180190 \alpha=180-190

We should note that it's not possible to get a negative result, and therefore there is no solution.

Answer

There is no possibility of resolving

Exercise #17

ABC is a Right triangle

Since BD is the median

and given that AC=10.

Find the length of the side BD.

AAABBBCCCDDD10

Video Solution

Step-by-Step Solution

We can calculate BD according to the following rule:

In a right triangle, the midpoint of the hypotenuse is equal to half of the hypotenuse.

That is:

BD is equal to half of AC:

Given that: AC=10 AC=10

BD=10:2=5 BD=10:2=5

Answer

5

Exercise #18

Find the measure of the angle α \alpha

898989AAABBBCCC

Video Solution

Step-by-Step Solution

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

Therefore, we will use the following formula:

A+B+C=180 A+B+C=180

Now let's input the known data:

89+512+α=180 89+5\frac{1}{2}+\alpha=180

α+94.5=180 \alpha+94.5=180

We'll move the term to the other side and keep the appropriate sign:

α=18094.5 \alpha=180-94.5

α=85.5 \alpha=85.5

Answer

8512 85\frac{1}{2}

Exercise #19

Find the measure of the angle α \alpha

49.649.649.6AAABBBCCC38

Video Solution

Step-by-Step Solution

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

Therefore, we will use the following formula:

A+B+C=180 A+B+C=180

Now let's input the known data:

α+49.6+38=180 \alpha+49.6+38=180

α+87.6=180 \alpha+87.6=180

We'll move the term to the other side and keep the appropriate sign:

α=18087.6 \alpha=180-87.6

α=92.4 \alpha=92.4

Answer

92.4

Exercise #20

Calculate the value of X.

11511511520+X20+X20+XAAABBBCCC8X

Video Solution

Step-by-Step Solution

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

Therefore, we will use the following formula:

A+B+C=180 A+B+C=180

We'll substitute the known data:

115+8x+20+x=180 115+8x+20+x=180

We'll combine similar terms:

9x+135=180 9x+135=180

We'll move terms to one side and maintain the appropriate sign:

9x=180135 9x=180-135

9x=45 9x=45

We'll divide both sides by 9:

x=5 x=5

Answer

5