## Sum and Difference of Angles

We can add angles and get the result of their sum, and we can also subtract them to find the difference between them.
Even if the angles don't have any numbers, we'll learn how to represent their sum or difference and arrive at the correct result.

## Angle Sum

To find the sum of angles, they must have a common vertex.

## Difference Between Angles

Just as we have added angles, we can also subtract one from another.

We can say that:

$∡BAE+∡EAC=∡BAC$

## Examples with solutions for Sum and Difference of Angles

### Exercise #1

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 #2

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 #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

In a right triangle, the sum of the two non-right angles is...?

### Step-by-Step Solution

In a right-angled triangle, there is one angle that equals 90 degrees, and the other two angles sum up to 180 degrees (sum of angles in a triangle)

Therefore, the sum of the two non-right angles is 90 degrees

$90+90=180$

90 degrees

### Exercise #5

What is the value of the void angle?

### Step-by-Step Solution

The empty angle is an angle adjacent to 160 degrees.

Remember that the sum of adjacent angles is 180 degrees.

Therefore, the value of the empty angle will be:

$180-160=20$

20

### Exercise #6

Calculate the size of angle X given that the triangle is equilateral.

### 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:

$x+x+x=180$

$3x=180$

We divide both sides by 3:

$x=60$

60

### Exercise #7

What type of angle is $\alpha$?



### Step-by-Step Solution

Let's remember that an acute angle is smaller than 90 degrees, an obtuse angle is larger than 90 degrees, and a straight angle equals 180 degrees.

Since in the drawing we have lines perpendicular to each other, the marked angles are right angles, each equal to 90 degrees.

Straight

### Exercise #8

Indicates which angle is greater

### Step-by-Step Solution

In drawing A, we can see that the angle is an obtuse angle, meaning it is larger than 90 degrees:

While in drawing B, the angle is a right angle, meaning it equals 90 degrees:

Therefore, the larger angle appears in drawing A.

### Exercise #9

Indicates which angle is greater

### Step-by-Step Solution

Note that in drawing B, the two lines form a right angle, which is an angle of 90 degrees:

While the angle in drawing A is greater than 90 degrees:

Therefore, the angle in drawing A is larger.

### Exercise #10

Indicates which angle is greater

### Step-by-Step Solution

Note that in drawing A, the angle is more acute, meaning it's smaller:

While in drawing B, the angle is more obtuse, meaning it's larger:

### Exercise #11

Indicates which angle is greater

### Step-by-Step Solution

Note that in drawing A, the angle is a straight angle equal to 180 degrees:

While in drawing B, we are given a right angle, equal to 90 degrees:

Therefore, the angle in drawing A is larger.

### Exercise #12

Indicates which angle is greater

### Step-by-Step Solution

In drawing A, we can see that the angle is more closed:

While in drawing B, the angle is more open:

In other words, in drawing A the angle is more acute, while in drawing B the angle is more obtuse.

We'll remember that the more obtuse an angle is, the larger it is.

Therefore, the larger angle appears in drawing B.

### Exercise #13

Indicates which angle is greater

### Step-by-Step Solution

Answer B is correct because the more closed the angle is, the more acute it is (less than 90 degrees), meaning it's smaller.

The more open the angle is, the more obtuse it is (greater than 90 degrees), meaning it's larger.

### Exercise #14

Can a triangle have more than one obtuse angle?

### Step-by-Step Solution

If we try to draw two obtuse angles and connect them to form a triangle (i.e., only 3 sides), we will see that it is not possible.

No

### Exercise #15

What is the size of angle ABC?

### Step-by-Step Solution

In order to calculate the value of angle ABC, we must calculate the sum of all the given angles.

That is:

$ABC=60+50$

$ABC=110$