# What is the full form of SSS

### What does congruent mean?

#### Example:

Look at the stop signs. These 4 stop signs are to each other **congruent**.

They are shifted, rotated or mirrored to each other.

Any two flat figures (triangles, squares, circles, ...) are called **congruent to each other**if you can transform them into one another by moving, rotating or mirroring them.

Displacements, rotations and reflections are therefore also called **Congruence maps**.

Congruence comes from the Latin word "congruentia" and means "congruence" in German.

### And what is not congruent?

#### Example:

These stop signs are not congruent because they have been enlarged or reduced:

Figures that are no longer congruent, but emerge from each other by enlarging or reducing the size, are called **similar**.

### Congruent triangles

If 2 triangles are congruent, then all of their sides and all angles match.

How can you quickly check if triangles are congruent to each other? For this you take one of the four **Congruence clauses**.

Here comes the first:

**The congruence theorem SSS (side - side - side)**

If 2 triangles match on all of their sides (S), they are congruent to each other.

The triangles can be rotated or mirrored:

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### Apply SSS

##### Example 1:

Triangle 1:

a = 4.5 cm, b = 3.8 cm, c = 2 cm

Triangle 2:

a = 4.5 cm, b = 3.8 cm, c = 2 cm

Obviously, triangle 1 and triangle 2 are now congruent to one another according to the congruence theorem SSS, because they agree on all three sides.

##### Example 2:

Triangle 3:

a = 4.5 cm, b = 3.8 cm, c = 2 cm

Triangle 4:

a = 2 cm, b = 4.5 cm, c = 3.8 cm

Triangle 3 and triangle 4 are now also congruent to one another according to the congruence theorem SSS. They agree on all three sides. However, here side a of triangle 3 corresponds to side b of triangle 4, side b of triangle 3 to side c of triangle 4, etc. The order of the sides is still the same.

As a reminder, in a triangle, the points are labeled A, B, and C in a counterclockwise direction, and the sides are labeled a, b, and c. The side is opposite to point A, side b to point B and side c to point C.

### Apply SSS

##### Example 3:

Triangle 5:

a = 4.5 cm, b = 3.8 cm, c = 2 cm

Triangle 6:

a = 4.5 cm, b = 2 cm, c = 3.8 cm

Now only sides b and c have been swapped in terms of their sizes, but the theorem is still applicable, triangles 5 and 6 are still congruent, but mirrored.

##### Example 4:

Triangle 7:

a = 4.5 cm, b = 3.8 cm, c = 2.1 cm

Triangle 8:

a = 4.5 cm, b = 2 cm, c = 3.8 cm

Side c of triangle 7 has no equivalent in triangle 8, the congruence theorem is not applicable and the two triangles are therefore not congruent to each other.

### Construct with the congruence theorem SSS

A triangle is exactly determined when all 3 sides are given. That means you can construct it with a compass and ruler.

In the following you should construct a triangle with the sides a = 5 cm, b = 3 cm and c = 7 cm. You do this as follows.

### Step 1:

Draw side c with vertices A and B horizontally.

### 2nd step:

Draw a circle $$ K_1 $$ with radius b around point A.

### 3rd step:

Draw a circle $$ K_2 $$ with radius a around point B.

### 4th step .:

Designate the intersection of the two circles $$ K_1 $$ and $$ K_2 $$ above side c with C.

### 5th step:

Connect points A and C to segment b and B and C to segment a and thereby complete the triangle.

**Note:** If you had chosen the lower and not the upper intersection point in step 4, you would have received a congruent triangle, but the counter-clockwise order of the points would no longer have been correct.

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### Proof of the congruence theorem SSS

### What is a sentence actually?

In mathematics one understands by one **sentence** a statement that is always valid. There is at least one proof for every proposition. The proof generally shows that the statement is always valid.

##### Example:

“The sum of the interior angles in a triangle is always 180 °. "

This is the so-called interior angle sum**sentence**.

Strictly speaking, you first have to prove the congruence theorem SSS in order to convince yourself that it is really valid. At least that's how a real mathematician would proceed. :-)

### The proof

You start from any triangle with sides a, b and c. If you start with side c, there are only two triangles: The intersection of the two circles is at the top **or** below. They match in all three lengths. These two triangles are congruent to each other because they have only been mirrored.

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