Parallel lines play a fundamental role in geometry, engineering and many other important fields. Learning to work with parallel lines will allow you to solve many different types of geometry problems at various levels of difficulty.

Properties of parallel lines

We can state the following about parallel lines:

Parallel lines are always coplanar.

The distance between two parallel lines is constant (never changes), meaning that they will never intersect.

We can also find parallel lines in quadrilaterals that have sides, like the following:

In parallelograms, rectangles, squares and rhombuses there are two pairs of parallel sides.

In trapezoids there is only one pair of parallel sides.

If you are interested in learning more about angles, try visiting one of the following articles:

When two straight lines intersect, four angles are formed. In the following image two straight lines $c$ and $d$ intersect, resulting in angles $1, 2, 3, 4$.

Opposite angles

Opposite angles are two anglesdirectly opposite eachother across the vertex (where the two lines intersect).

The intersection of two straight lines results in two pairs of opposite angles.

Opposite angles are non-adjacent, meaning that they can not be two angles that are next to eachother.

Adjacent angles are two angles formed by the intersection of two lines (or rays).

Adjacent angles share a side.

Two adjacent angles are supplementary, i.e., the sum of their values is equal to $180º$.

In the following figure :

1 and 2 are adjacent angles

2 and 3 are adjacent angles

3 and 4 are adjacent angles

4 and 1 are adjacent angles

We can therefore state that:

$\sphericalangle1+\sphericalangle2=180°$

$\sphericalangle2+\sphericalangle3=180°$

$\sphericalangle3+\sphericalangle4=180°$

$\sphericalangle4+\sphericalangle1=180°$

Angles formed by a transversal

A line that intersects two parallel lines at different points is called a transversal. When a transversal intersects two parallel lines, eight angles are formed, four at each point of intersection. In the following picture, two parallel lines l and m are intersected by transversal line s. Eight angles 1, 2, 3, 4, 5, 6, 7 and 8 are formed.

Figure 3 :

Classification of angles

Depending on their position, the angles formed can either be:

Internal angles: These are the angles that are in between the two parallel lines.

In Figure 3 angles 3, 4, 5 and 6 are internal angles.

OR

External angles: These are the angles that are not in between the parallel lines.

In Figure 3 angles 1, 2, 7 and 8 are external angles.

Two angles formed by a transversal intersecting two parallel lines can be alternate angles, conjugate angles or corresponding angles, depending on which parts of the transversal forms those angles.

In the following image the angles $α$ y $ß$ are corresponding angles

Two corresponding angles are on the same side of the transversal line.

One of the corresponding angles will be an external angle while the other will be an internal angle.

Two corresponding angles do not share any of their sides.

In Figure:

Angles 1 and 5 are corresponding

Angles 2 and 6 are corresponding

Angles 3 and 7 are corresponding

Angles 4 and 8 are corresponding

We can state that:

If two parallel straight lines are cut by a transversal, then the corresponding angles are equal.

Which means that in figure 3:

$\sphericalangle1=\sphericalangle5$

$\sphericalangle2=\sphericalangle6$

$\sphericalangle3=\sphericalangle7$

$\sphericalangle4=\sphericalangle8$

Parallel lines practice problems

Exercise 1: parallel lines

In the following image, be $a||b$

Question:

What is the value of $ß$?

Solution:

We can see that the angles $α$ y $ß$ are corresponding angles. We know that when two parallel lines like $a$ and $b$ are cut by a transversal like $c$, the corresponding angles are equal and, therefore $ß=40º$

Here we have two parallel lines cut by a transversal. Since we know that angle $ß$ and the angle marked $130º$ are corresponding angles, then we know that these angles are equal and therefore. $ß=130º$.

Now we have to find the value for angle $∡α$ . Since the angles $∡α$ and $∡ß$ are adjacent, then we know that they are supplementary, which means that they add up to $180º$. Therefore,

$α+ß=180º$

By replacing$ß$ with its value we get the following:

$α+130º=180º$

Subtracting it results in

$α = 50º$

Exercise 3: parallel lines

How many parallel lines are there in the following graph?

Explanation

In the graph you can see:

that the straight line $f$ intersects the straight lines $b$ and $c$ (in dashed lines) at two points

that at both points of intersection the angle of intersection is the same $(90°)$

that these two angles are corresponding

Therefore the straight lines $b$ and $c$ are parallel.

In the following graph you can see

that the line $b$ intersects the lines $d$ and $e$ (in dashed lines) in two points

that at both points of intersection the angle of intersection is the same $(130°)$

that these two angles are external alternate angles

Therefore, it can be said that the straight lines $d$ and $e$ are parallel.

Solution:

Therefore, the final answer is that the graph has $2$ pairs of parallel lines.

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.

Answer

B

Exercise #2

Which lines are perpendicular to each other?

Video Solution

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.

Answer

Exercise #3

What can be said about the lines shown below?

Video Solution

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.

Therefore, no answer is correct.

Answer

None of the above.

Exercise #4

What do the four figures below have in common?

Video Solution

Step-by-Step Solution

Answer

All parallel

Exercise #5

Which of the figures shows parallel lines?

Video Solution

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.