# Sum and difference of angles - Examples, Exercises and Solutions

## What is an angle?

Definition: Angles are created at the intersection between two lines. As seen in the following illustration

The angle in the illustration is called $AB$. We could also call it angle $\sphericalangle ABC$. The important thing is that the middle letter is the one at the intersection of the lines.

For example, in this case:

The angle is $\sphericalangle BCD$ or $\sphericalangle DCB$. Both notations are correct for the same angle.

We usually mark the angle with an arc as follows:

The marked angle is $∡ABC$. Sometimes we will denote angles using Greek letters, for example:

$α$ or $β$

Before the name of the angle, we should note the angle symbol, like this:

$∡$

Together it looks like this:

$∡CBA$ or $∡α$

Next, we will delve into the size of angles, the different types, and those that are created when a line intersects two parallel lines.

## 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 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 #3

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 #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

### Answer

$65$°

### 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.

### Answer

The triangle is incorrect.

### Exercise #1

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 #2

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.

### Answer

90°, right angle.

### Exercise #3

ABCD is a quadrilateral.

$∢A=80$

$∢C=95$

$∢D=45$

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 #4

ABCD is a quadrilateral.

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 #5

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 #1

ABCD is a quadrilateral.

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 #2

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°

### Exercise #3

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

90 degrees

### Exercise #4

What is the value of the void angle?

20

### Exercise #5

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

60