 The Thales Theorem is a ninth grade math course that is very important to a student. In this lesson we will calculate the length of a segment or otherwise demonstrate whether two lines are parallel or not. The student will need to know how to apply the direct and reciprocal part of the theorem and use the cross product rule.
We will end this lesson by solving real-life problems in the third grade.

## I. The direct part of the theorem:

### 1.the theorem of Thales :

Ownership:

If we have :

• ;
• • then we have the following equalities of ratios : .

Three configurations illustrate the theorem of Thales called “triangle”, “hourglass”, “eight”. Remark:

The lengths of triangle AMN are proportional to the lengths of triangle ABC.

### 2.length calculations :

Example: The figure above is composed of four straight lines.

The blue lines are parallel.

DG=25 mm, GH=45 mm, CG = 20 mm, HT = 27 mm.

The lines (DH) and (CT) are secant at G.

The lines (CD) and (HT) are parallel.

According to the direct part of Thales’ theorem, we have the following equalities: or .

GT calculation: CD calculation: ### 3.prove that two lines are not parallel:

Example: Above, the lines (ES) and (MR) are secant at T.

TR= 11 cm; TS = 8 cm; TM = 15 cm and TE= 10 cm.

On the one hand, .

It can be seen that .

Now, if the lines (RS) and (ME) were parallel, according to the theorem of Thales, there would be equality.

Since this is not the case, the lines (RS) and (ME) are not parallel.

## II. The reciprocal theorem:

### Reciprocal of Thales’ theorem:

Ownership:

If the points A,N,C are aligned in the same order andif we have : then . Remark:

Be careful, it is not enough to check the equality of the ratios: it is also necessary to ensure that the points are placed in the right order.

### 2.prove that two lines are parallel:

Example: Above, the lines (HA) and (TL) are secant at M.

On the one hand, , on the other hand .

It can be seen that .

Moreover, the points A,M,H on the one hand and the points M,L,T on the other hand are aligned in the same order.

Therefore, according to the reciprocal of Thales’ theorem, the lines (AL) and (HT) are parallel.

## Did you assimilate this lesson on this theorem in 9th grade?

Le théorème de Thalès

QCM sur le théorème de Thalès en 3ème.

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