# Angles in Parallel Lines - Examples, Exercises and Solutions

## Angles on Parallel Lines

If we add a third line that intersects the two parallel lines (those lines that could never cross), we will obtain various types of angles.
To classify these angles we must observe if they are:
above the line - the pink part
below the line - the light blue part
to the right of the line - the red part
to the left of the line - the green part

### Suggested Topics to Practice in Advance

1. Parallel lines

## Examples with solutions for Angles in Parallel Lines

### Exercise #1

Is it possible to have two adjacent angles, one of which is obtuse and the other right?

### Step-by-Step Solution

Remember the definition of adjacent angles:

Adjacent angles always complement each other up to one hundred eighty degrees, that is, their sum is 180 degrees.

This situation is impossible since a right angle equals 90 degrees, an obtuse angle is greater than 90 degrees.

Therefore, together their sum will be greater than 180 degrees.

No

### Exercise #2

In which of the diagrams are the angles $\alpha,\beta\text{ }$ vertically opposite?

### Step-by-Step Solution

Remember the definition of angles opposite by the vertex:

Angles opposite by the vertex are angles whose formation is possible when two lines cross, and they are formed at the point of intersection, one facing the other. The acute angles are equal in size.

The drawing in answer A corresponds to this definition.

### Exercise #3

Which type of angles are shown in the diagram?

### Step-by-Step Solution

Let's remember that corresponding angles can be defined as a pair of angles that can be found on the same side of a transversal line that intersects two parallel lines.

Additionally, these angles are positioned at the same level relative to the parallel line to which they belong.

Corresponding

### Exercise #4

Which type of angles are shown in the figure below?

### Step-by-Step Solution

Alternate angles are a pair of angles that can be found on the opposite side of a line that cuts two parallel lines.

Furthermore, these angles are located on the opposite level of the corresponding line that they belong to.

Alternate

### Exercise #5

What angles are described in the drawing?

### Step-by-Step Solution

Let's remember that vertical angles are angles that are formed when two lines intersect, and they are created at the point of intersection, opposite each other.

Vertices

### Exercise #6

$a$ is parallel to

$b$

Determine which of the statements is correct.

### Step-by-Step Solution

Let's review the definition of adjacent angles:

Adjacent angles are angles formed where there are two straight lines that intersect. These angles are formed at the point where the intersection occurs, one next to the other, and hence their name.

Now let's review the definition of collateral angles:

Two angles formed when two or more parallel lines are intersected by a third line. The collateral angles are on the same side of the intersecting line and even are at different heights in relation to the parallel line to which they are adjacent.

Therefore, answer C is correct for this definition.

### Answer

$\beta,\gamma$ Colaterales$\gamma,\delta$ Adjacent

### Exercise #7

Look at the rhombus in the figure.

What is the relationship between the marked angles?

### Step-by-Step Solution

Let's remember the different definitions of angles:

Corresponding angles are angles located on the same side of the line that intersects the two parallels and are also situated at the same level with respect to the parallel line they are adjacent to.

Therefore, according to this definition, these are the angles marked with the letter A

Alternate angles are angles located on two different sides of the line that intersects two parallels, and which are also not at the same level with respect to the parallel they are adjacent to.

Therefore, according to this definition, these are the angles marked with the letter B

### Answer

A - corresponding; B - alternate

### Exercise #8

Look at the rectangle ABCD below.

What type of angles are labeled with the letter A in the diagram?

What type are marked labeled B?

### Step-by-Step Solution

Let's remember the definition of corresponding angles:

Corresponding angles are angles located on the same side of the line that cuts through the two parallels and are also situated at the same level with respect to the parallel line to which they are adjacent.

It seems that according to this definition these are the angles marked with the letter A.

Let's remember the definition of adjacent angles:

Adjacent angles are angles whose formation is possible in a situation where there are two lines that cross each other.

These angles are formed at the point where the intersection occurs, one next to the other, and hence their name.

Adjacent angles always complement each other to one hundred eighty degrees, that is, their sum is 180 degrees.

It seems that according to this definition these are the angles marked with the letter B.

### Answer

A - corresponding

B - adjacent

### Exercise #9

What angles are described in the drawing?

### Step-by-Step Solution

Since we are not given any information about the lines, we cannot define the lines as parallel.

As a result, none of the options are correct.

### Answer

None of the possibilities

### Exercise #10

What angles are described in the drawing?

### Step-by-Step Solution

Since the angles are not on parallel lines, none of the answers are correct.

### Answer

Ninguna de las respuestas

### Exercise #11

The lines a and b are parallel.

What are the corresponding angles?

### Step-by-Step Solution

Given that line a is parallel to line b, let us remind ourselves of the definition of corresponding angles between parallel lines:

Corresponding angles are angles located on the same side of the line that intersects the two parallels and are also situated at the same level with respect to the parallel line to which they are adjacent.

Corresponding angles are equal in size.

According to this definition $\alpha=\beta$and as such they are the corresponding angles.

### Answer

$\alpha,\beta$

### Exercise #12

a is parallel to b.

Calculate the angles shown in the diagram.

### Step-by-Step Solution

Given that according to the definition, the vertex angles are equal to each other, it can be argued that:

$115=2$Now we can calculate the second pair of vertex angles in the same circle:

$1=3$

Since the sum of a plane angle is 180 degrees, angle 1 and angle 3 are complementary to 180 degrees and equal to 65 degrees.

We now notice that between the parallel lines there are corresponding and equal angles, and they are:

$115=4$

Since angle 4 is opposite to angle 6, it is equal to it and also equal to 65 degrees.

Another pair of alternate angles are angle 1 and angle 5.

We have proven that:$1=3=65$

Therefore, angle 5 is also equal to 65 degrees.

Since angle 7 is opposite to angle 5, it is equal to it and also equal to 115 degrees.

That is:

$115=2=4=6$

$65=1=3=5=7$

### Answer

1, 3 , 5, 7 = 65°; 2, 4 , 6 = 115°

### Exercise #13

What is the value of X given that the angles are between parallel lines?

### Step-by-Step Solution

The angle X given to us in the drawing corresponds to an angle that is adjacent to an angle equal to 154 degrees. Therefore, we will mark it with an X.

Now we can calculate:

$x+154=180$

$x=180-154=26$

26°

### Exercise #14

Calculates the size of the angle $\alpha$

### Step-by-Step Solution

Let's review the definition of alternate angles between parallel lines:

Alternate angles are angles located on two different sides of the line that intersects two parallels, and that are also not at the same level with respect to the parallel they are adjacent to. Alternate angles have the same value as each other.

Therefore:

$\alpha=40$

### Answer

$40$

### Exercise #15

Given two parallel lines

Calculate the angle $\alpha$

### Step-by-Step Solution

The angle 125 and the angle alpha are vertically opposite angles, so they are equal to each other.

$\alpha=125$

### Answer

$125$