Parallel lines

🏆Practice parallel lines

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 a and b b as shown below.

What are parallel lines

If we state the following:

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

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

a b a\parallel~b

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Test yourself on parallel lines!

einstein

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

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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:

On the Tutorela blog you will find a variety of interesting articles about mathematics.


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Angles formed by intersecting lines

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

Image Angles that result when two reticles are cut together


Opposite angles

1a - Angles opposite by the vertex
  • Opposite angles are two angles directly 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.
  • Opposite angles are equal.

In the following figure:

Image Angles that result when two reticles are cut together
  • 1 and 3 are opposite angles.
  • 2 and 4 are opposite angles.

We can therefore affirm that:

  • 1=3 \sphericalangle1=\sphericalangle3
  • 2=4 \sphericalangle2=\sphericalangle4

Do you know what the answer is?

Adjacent angles

  • 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 180º .

In the following figure :

Image Angles that result when two reticles are cut together

  • 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:

  • 1+2=180° \sphericalangle1+\sphericalangle2=180°
  • 2+3=180° \sphericalangle2+\sphericalangle3=180°
  • 3+4=180° \sphericalangle3+\sphericalangle4=180°
  • 4+1=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 :

Figure 2 Angles that result when two parallel straight lines

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.


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Alternate angles

  • Two angles are alternate angles if they are on opposite sides of the transversal line.
  • Two alternate angles can either be both external angles or both internal angles.
  • Two alternate angles do not share any of their sides.

In Figure 3:

  • Angles 4 and 6 are internal alternates.
  • Angles 3 and 5 are internal alternates.
  • Angles 1 and 7 are external alternates.
  • Angles 2 and 8 are external alternates.

In the following image we can see two pairs of internal alternate angles, one highlighted in red and the other in blue.

A - you can see two pairs of alternating internal angles

We can state that:

If two parallel lines are cut by a transversal, then the pairs of internal alternate angles are equal.

If two parallel lines are cut by a transversal, then the pairs of alternate exterior angles are equal.

Which means that in Figure 3:

  • 4=6 \sphericalangle4=\sphericalangle6
  • 3=5 \sphericalangle3=\sphericalangle5
  • 1=7 \sphericalangle1=\sphericalangle7
  • 2=9 \sphericalangle2=\sphericalangle9

Conjugate angles

  • Conjugate angles are on the same side of the transversal line.
  • Conjugate angles can be both external or both internal.

In Figure 3:

  • Angles 3 and 6 are internal conjugates.
  • Angles 4 and 5 are internal conjugates.
  • Angles 2 and 7 are external conjugates.
  • Angles 1 and 8 are external conjugates.
C - Alternate angles

We can state that:

If two parallel lines are cut by a transversal then the pairs of internal conjugate angles are supplementary, i.e. their sum equals 180 degrees.

If two parallel lines are cut by a transversal, then the pairs of external conjugate angles are supplementary.

Which means that in Fig:

Figure 2 Angles that result when two parallel straight lines
  • 3+6=180° \sphericalangle3+\sphericalangle6=180°
  • 4+5=180° \sphericalangle4+\sphericalangle5=180°
  • 2+7=180° \sphericalangle2+\sphericalangle7=180°
  • 1+9=180° \sphericalangle1+\sphericalangle9=180°

Do you think you will be able to solve it?

Corresponding angles

In the following image the angles α α y ß 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:

Figure 2 Angles that result when two parallel straight lines
  • 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:

  • 1=5 \sphericalangle1=\sphericalangle5
  • 2=6 \sphericalangle2=\sphericalangle6
  • 3=7 \sphericalangle3=\sphericalangle7
  • 4=8 \sphericalangle4=\sphericalangle8

Parallel lines practice problems

Exercise 1: parallel lines

In the following image, be ab a||b

Question:

What is the value of ß ß ?

Exercise 1 on parallel lines

Solution:

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


Test your knowledge

Exercise 2: parallel lines

In the following image ab a||b

Question:

What are the values of α α and ß ß ?

AB A\Vert B

Observe the plane and solve:

  • ß=? ß=?
  • α=? α=?
1a - Exercise 2- on parallel lines

Solution:

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

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

α+ß=180º α+ß=180º

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

α+130º=180º α+130º=180º

Subtracting it results in

α=50º α = 50º


Exercise 3: parallel lines

How many parallel lines are there in the following graph?

Exercise 3 on parallel lines

Explanation

In the graph you can see:

  • that the straight line f f intersects the straight lines b b and c c (in dashed lines) at two points
  • that at both points of intersection the angle of intersection is the same (90°) (90°)
  • that these two angles are corresponding

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

Exercise 3 on parallel straight lines elaborated

In the following graph you can see

  • that the line b b intersects the lines d d and e e (in dashed lines) in two points
  • that at both points of intersection the angle of intersection is the same (130°) (130°)
  • that these two angles are external alternate angles

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

exercise 3 second part

Solution:

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


Do you know what the answer is?

Exercise 4: parallel lines

How many degrees do we have to add to angle β β so that there will be another parallel line in the following graph?

Exercise 3 on parallel lines

Explanation

By adding 4° degrees to angle β ∡βwe will get an angle of 90° 90° degrees, and by doing so we will create another line parallel to the two below it.

86°+4°=90° 86°+4°=90°

Exercise 4 on parallel lines solution

Solution:

The correct answer is:


Exercise 5: parallel lines

This question is divided into several parts:

  1. How many degrees is angle ABC ∡ABC and what kind of angle is it in relation to CBF ∡CBF ?
  2. How many degrees is angle BDE ∡BDE and what kind of angle is it in relation to ADC ∡ADC ?
Exercise 5 on parallel lines

Answer 1:

A. Angle ABC ∡ABC is equal to 180º130º=50º 180º-130º=50º

B. Angle ABC ∡ABC is adjacent to angle CBF ∡CBF .

Answer 2:

  1. Angle BDE ∡BDE is equal to 90º 90º because it is the opposite angle of angle ADC=90º ∡ADC=90º

Check your understanding

examples with solutions for parallel lines

Exercise #1

Which of the diagrams contain parallel lines?

AB

Video Solution

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.

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?

1234

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.

Answer

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