# Parallel lines - Examples, Exercises and Solutions

## What are parallel lines?

Parallel lines are lines that belong to the same plane (are coplanar) and never meet (do not intersect).

Let there be two parallel lines $a$ and $b$ as shown below.

If we state the following:

The straight line $a$ is parallel to the straight line $b$

we can say the same thing using mathematical language as follows:

$a\parallel~b$

### Suggested Topics to Practice in Advance

1. Perpendicular Lines

## examples with solutions for parallel lines

### Exercise #1

Which of the diagrams contain parallel lines?

### Step-by-Step Solution

In drawing B, we observe two right angles, which teaches us that they are practically equal. From this, we can conclude that they are corresponding angles, located at the intersection of two parallel lines.

In drawing A, we only see one right angle, so we cannot deduce that the two lines are parallel.

B

### Exercise #2

Which lines are perpendicular to each other?

### Step-by-Step Solution

Let's remember that perpendicular lines are lines that form a right angle of 90 degrees between them.

The only drawing where it can be seen that the lines form a right angle of 90 degrees between them is drawing A.

### Exercise #3

What can be said about the lines shown below?

### Step-by-Step Solution

Let's remember the different properties of lines.

The lines are not parallel since they intersect.

The lines are not perpendicular since they do not form a right angle of 90 degrees between them.

None of the above.

### Exercise #4

What do the four figures below have in common?

All parallel

### Exercise #5

Which of the figures shows parallel lines?

### Step-by-Step Solution

Parallel lines are lines that, if extended, will never meet.

In the drawings A+B+D if we extend the lines we will see that at a certain point they come together.

In drawing C, the lines will never meet, therefore they are parallel lines.

## examples with solutions for parallel lines

### Exercise #1

Which figure shows perpendicular lines?

### Step-by-Step Solution

Perpendicular lines are lines that form a right angle between them.

In the drawings A+C+D, you can see that the angles formed are not right angles.

It is possible to point out a right angle in drawing B.

### Exercise #2

Which of the figures show perpendicular lines?

### Step-by-Step Solution

Perpendicular lines are lines that form a right angle of 90 degrees between them.

It can be observed that in figures 1 and 3, the angles formed by the lines between them are right angles of 90 degrees.

1 and 3

### Exercise #3

Which lines are perpendicular to each other?

### Step-by-Step Solution

Perpendicular lines are lines that form a right angle of 90 degrees between them.

The only drawing where the lines form a right angle of 90 degrees between them is drawing A.

### Exercise #4

The lines below are not the same size, but are they parallel?

### Step-by-Step Solution

Remember the properties of parallel lines.

Since there is no connection between the size of the line and parallelism, the lines are indeed parallel.

Yes

### Exercise #5

Are lines AB and DC parallel?

### Step-by-Step Solution

For the lines to be parallel, the two angles must be equal (according to the definition of corresponding angles).

Let's compare the angles:

$2x+10=70-x$

$2x+x=70-10$

$3x=60$

$x=20$

Once we have worked out the variable, we substitute it into both expressions to work out how much each angle is worth.

First, substitute it into the first angle:

$2x+10=2\times20+10$

$40+10=50$

Then into the other one:

$70-20=50$

We find that the angles are equal and, therefore, the lines are parallel.

Yes

## examples with solutions for parallel lines

### Exercise #1

Given: $3\alpha=x$

Are they parallel lines?

### Step-by-Step Solution

If the lines are parallel, the two angles will be equal to each other, since alternate angles between parallel lines are equal to each other.

We will check if the angles are equal by substituting the value of X:

$x+\alpha+31=3\alpha+\alpha+31=4\alpha+31$

Now we will compare the angles:

$4\alpha+31=4\alpha+29$

We will reduce on both sides to$4\alpha$We obtain:
$31=29$

Since this theorem is not true, the angles are not equal and, therefore, the lines are not parallel.

No

### Exercise #2

Which figure(s) show intersecting lines?

1 and 3

### Exercise #3

What do the 4 figures below have in common?

### Video Solution

All intersections

### Exercise #4

What do the four figures below have in common?

### Video Solution

All perpendicular

### Exercise #5

What do the four figures below have in common?