Everything you need to know about Thales’ theorem

Along with the Pythagorean theorem, Thales’ theorem is one of the basics of geometry that a third grader should master. There is a good chance that exercises on this mathematical theorem will be asked in the college certificate!

Its mastery is also required to pass the Tage Mage and Score IAE Message test of certain business schools.

So, it is better to understand what it is, why it is and how to use it to successfully complete the exercises. We explain everything in this article!

Definition of Thales theorem in mathematics

the Thales theorem is a mathematical theory proposed by a Greek philosopher and scientist named Thales of Miletus 500 years before Christ.

He first used this theorem to determine the properties of the Cheops pyramid in Egypt. Class, huh?

the Thales theorem claims that a line parallel to one of the sides of a triangle defines a second triangle with proportional angles using the other two sides.

What is the current Thales theorem for?

It’s all good to know the definition of a theorem, but it’s even better to know what it’s used for.

you will use in math exercises to calculate length within certain geometric figures.

It is used to demonstrate that opposite lines are parallel or not opposite.

What do you need to use Thales theorem?

When to apply Thales theorem?

To be able to use the theorem, you need to be next to the geometric figure that corresponds to it two parallel lines intersected by two intersecting lines.

The theorem can be applied only in two cases (see the diagram below):

  • Two intersecting lines and two parallel lines form two distinct triangles connected by a vertex.
  • Two intersecting lines and two parallel lines form two nested triangles with a common vertex

As a reminder:

  • Two lines are parallel if they have no points in common. They are different and will never cross.
  • Two straight lines intersect at a point called a point of intersection.

You also need to know at least three lengths in this type of figure to be able to use it Thales theorem.

What is the formula for Thales’ theorem?

An example of Thales’ theorem

In a math exercise, you will be presented with a statement asking you to calculate the length within a geometric figure and number.

For example :

Statement: Calculate the length of AC and subtract the length of AB.

Step one: state the theorem.

Consider triangle ACE and two points B on line (AC) and D on line B and (EC) such that line (BD) is parallel to line (AE) as shown in the figure above. .

According to Thales theorem:


Step two: then you need to replace the lengths with the known measurements.

2/AC = DC/EC = 1/5

2/AC = 1/5

AC = (2 × 5) / 1
AC = 10/1
AC = 10

Last step: based on the previously found result, you need to deduce the length of segment AB.

AB = CA – CB
AB = 10 – 2
AB = 8

So CA measures 10 cm and AB measures 8 cm.

Make way for Thales theorem exercise: apply the formula

Statement of the problem:

Consider the figure below:

using Thales theoremCalculate the length of segment HI to the nearest 0.1 centimeter.


Let the two triangles be GFJ and GHI. We know that lines (FJ) and (HI) are parallel.

according to Thales theorem :


We replace:

GJ/GH = 7/8 = 6/HI

7/8 = 6/HI

HI = (6 × 8) / 7
HI = 48/7
HI = 6.9

So HI is about 6.9 cm.

How do you show that two lines are parallel? The equivalent of Thales

to use The converse of Thales’ theorem, you should know four measurements. It is used to prove or disprove that two intersecting straight lines are parallel.

Consider the figure below:

You are asked to prove that (BE) is parallel to (CD). For this we will first calculate AE/AD and then calculate BE/CD.

AE/AD = 3/(3+6) = 3/9 = 1/3

BE/CD = 2/6 = 1/3

Thus, AE/AD = BE/CD, by the converse of Thales’ theorem, two straight lines are parallel.

If the results obtained after the calculation are different, it means that the two straight lines are not parallel.

Make way for an exercise on the converse of Thales’ theorem

Consider the figure below:

We know the following measures:

DA=4cm, AC=10cm, DE=3cm, BC=7cm

Are segments DE and BC parallel?

And here is the answer with a suitable demonstration:

AD/AC = 4/10 = 2/5

DE/BC = 3/7

To be safe, you can cross-product:

3 × 10 = 30

The results are therefore not equal The converse of Thales’ theoremThe segments DE and BC are not parallel.

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