To understand Thales’ theorem you need to understand very well what is the reason between two segments.

For example, we have these two segments:

As you know, the segments are delimited by two extremes and are named by the extremes that limit it. The red segment, which begins at end A and ends at end B, is called segment AB.

If two segments have some relation between them, the same letters are used to name them, but since they cannot be repeated, the single quotation mark next to each letter is used and the quotation mark reads “prima”. So A’ would read “A prima”.

Thus, the blue segment, which starts at A’ and ends at B’, will be called A’ B’ segment.

Box]The ratio (or ratio) of two segments is the result of dividing the length of those two segments.

If the AB segment measures 5 cm and the A’ B’ segment measures 10 cm, what is the reason for these two segments?

All we have to do is divide the length of segment AB by the length of segment A’ B’:

The ratio of these two segments is 0.5, which means that AB is half that of A’ B’.

We can also calculate the ratio by dividing the length of segment A’ B’ by the length of segment AB:

In this case, the ratio is 2, or in other words, segment A’ B’ is twice as high as segment AB.

If you notice, to say that the AB segment is half as much as the A’ B’ segment, is the same as saying that the A’ B’ segment is twice as much as the AB segment.

Therefore, it is not necessary to calculate the ratio in both ways. Calculating it in one of two ways is enough.

Proportionality between pairs of segments

We now have these two segments:

The red segment, CD, measures 3 cm and the blue segment C’ D’ measures 6 cm.

Let’s figure out his reason:

The ratio of the CD and C’ D’ segments is the same as the ratio of the AB and A’ B’ segments.

When two pairs of segments have the same reason, they are said to be proportional.

Therefore, segments AB and A’ B’ are proportional to CD and C’ D’:

Two pairs of segments are proportional when their ratio is the same

Thales’ theorem

Once I’ve explained the reasoning between two segments and the proportionality between two pairs of segments, let’s look at Thales’ theorem.

We have two secant (non-parallel) straight lines. One is called the straight line r (red color) and the other is called the straight line s (blue color):

To these two straight lines, we cut them with several parallel lines (green color), as follows:

To the points where they cut the lines parallel to the straight line, I will call them A, B and C and to the points where they cut the lines parallel to the straight line, I will call them A’, B’ and C’:

The green lines have divided the line r into two segments: segment AB and segment BC. We also have a third segment if we consider the first and last parallel line, i. e. the AC segment.

They have also divided the line s into two segments A’ B’ and B’ C’ and if we consider the first and last parallel line, there is a third segment A’ C’.

Thales’ theorem tells us the following:

Box]When any two straight lines, r and s, are cut by several parallel lines, the segments that make up the line r are proportional to the segments that make up the line s.

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And what does that mean?

So if you divide the lengths of the segments that are at loggerheads, i. e. segment AB by segment A’ B’ they have the same reason as if you divide segment BC by segment B’ C’:

As they have the same reason, AB and A’ B’ are proportional to BC and B’ C’.

If we consider the segment formed by the first and last parallel line, i. e. the AC segment, it is also proportional to the AB segment:

And therefore, all segments of the line r are proportional to the segments of the line s:

What is Thales’ theorem for?

Thales’ theorem allows you to calculate the length of a segment, knowing the values of all the other segments of two straight lines that are in Thales’ position.

To be in Thales’ position means that the straight lines must be as the Thales theorem says, that is, two straight lines cut by several parallel straight lines.

We’ll solve several exercises to make it much clearer.

Thales’s Theorem Solved Exercises

Exercise 1

The lines a and b of the drawing are parallel. Check using Thales’s theorem if so is the line c.

How do we prove that straight line c is parallel?

For we have to prove that the straight lines are in Thales’s position and that Thales’ theorem is fulfilled, checking if the segments of both lines have the same reason and that between them they are proportional.

We calculate the reason for the first segments:

And the reason for the next two segments:

The reason is the same, so both pairs of segments are proportional.

Then the theorem of Thales is fulfilled and as a consequence, the straight line c is parallel.

Exercise 2

How much does the x-segment in this drawing measure?

We know what the two segments of r measure, but it remains to know how much one of the segments of s measures, so we call that segment x.

Then, according to Thales’ theorem, the sections that are at loggerheads have the same reason, so their divisions should be the same and therefore we can match them:

We have one first-degree equation left, from where we have to clear the x.

There is much confusion to solve equations of this kind.

To solve this equation, we pass the denominators of each member, multiplying the numerator of the opposite member (multiply in cross).

The 5 that is dividing the 8 in the first member, passes multiplying the 6 in the second member and the x, which is dividing the 6 in the second member, passes multiplying the 8 in the first member and we are left like this:

We no longer have denominators. Let’s clear the x.

Now, the 8 that is multiplying x, passes to the second member dividing:

And finally we operate to calculate the value of x:

If you check, the segment pairs will be proportional.