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## SSS theorem definition

The triangles having the same shape and size are **congruent triangles**.

That is the triangles have corresponding angles and corresponding sides. We can test its congruence using some theorems without checking all the angles and sides of triangles. And one of the theorems is the SSS theorem.

**SSS theorem** : If all the three corresponding sides of two triangles are equal to each other, then they are congruent.

So as the name suggests, this theorem stands for Side-Side-Side. Here we only take a look at the sides of the triangle and not anything else.

## SSS congruence theorem

The SSS congruence theorem gives the congruence relation between two triangles based on their sides.

**SSS congruence theorem** : The two triangles are congruent if all the three respective sides of both the triangles are equal.

Mathematically, if $AB=XY,BC=YZ,$ and id="2618600" role="math" $AC=XZ$, then $\u25b3ABC\cong \u25b3XYZ.$

So if we can replace all the three sides of one triangle with all the sides of another triangle then both triangles are congruent using the SSS criterion. In this situation, both triangles are represented with a congruency symbol.

As it is given we know that all three sides of both the triangles $\u25b3ABC$ and $\u25b3XYZ$ are of the same size and same length with each other. So we can lay sides XY on AB, YZ on BC, and XZ on AC by superimposing both the triangles. Hence that gives that $AB=XY,BC=YZ,AC=XZ.$ So $\u25b3ABC\cong \u25b3XYZ.$

### SSS congruence triangle examples

Here we will see some examples of SSS congruence to understand it.

Show that the given triangles are congruent to each other.

**Solution:**

We can see from the figure $AB=DE=7,BC=EF=11,AC=DF=15.$ As all the three sides both the triangles are equal to each other respectively, we can directly use the SSS congruence theorem.

Hence, $\u25b3ABC\cong \u25b3DEF.$

## SSS similarity theorem

In triangles if the corresponding angles are congruent and corresponding sides are proportional then both the triangles are similar. But to check this we don’t necessarily have to consider all the sides and angles. We can simply use the SSS similarity theorem and the knowledge of Proportional sides to prove similar triangles.

**SSS Similarity Theorem** : Two triangles are said to be similar when the corresponding sides of these two triangles are proportional.

Proof: We are given that the corresponding sides of two triangles are proportional.

That is, $\frac{AB}{MN}=\frac{BC}{NO}=\frac{AC}{MO}\left(1\right)$

To prove: $\u25b3ABC~\u25b3MNO$

First, we consider two points P and Q on lines MN and MO respectively such that $MP=AB$ and $MQ=AC$. Now we join these points and form a line PQ such that PQ is parallel to NO.

We can construct line PQ by parallel postulate, which states that there is one parallel line passing through any point not on that line in the same plane.

Then we substitute AB and AC with MP and MQ respectively in equation 1.

$\Rightarrow \frac{MP}{MN}=\frac{MQ}{MO}$

Now, as $PQ\parallel NO,\angle MPQ=\angle N$ and $\angle MQP=\angle O$ are corresponding angles respectively. Hence by applying AA - Similarity we have $\u25b3MPQ~\u25b3MNO.$

From the definition of similar triangles on $\u25b3MPQ$ and $\u25b3MNO,$ we get that

$\frac{MP}{MN}=\frac{MQ}{MO}=\frac{PQ}{NO}\left(2\right)$

Again substituting id="2618772" role="math" $MP=AB$ and id="2618771" role="math" $MQ=AC$ in equation 1, we get

$\frac{MP}{MN}=\frac{BC}{NO}=\frac{MQ}{MO}\left(3\right)$

So comparing equation 2 and equation 3 $\frac{PQ}{NO}=\frac{BC}{NO}\Rightarrow PQ=BC.$

Finally, we know that id="2618781" role="math" $AB=MP,BC=PQ,AC=MQ$. So by the SSS congruence theorem, we get id="2618782" role="math" $\u25b3ABC\cong \u25b3MPQ.$And we also have that id="2618785" role="math" $\u25b3MPQ~\u25b3MNO.$ Hence from both the similarity we get id="2618788" role="math" $\u25b3ABC~\u25b3MNO.$

## SSS similarity theorem examples

Let us take a look at SSS similarity theorem examples.

Check if the given triangles are similar or not.

**Solution:**

Here to determine similar triangles we need to check the proportional sides. So first we will find the ratios of the corresponding sides.

$\Rightarrow \frac{DE}{AB}=\frac{4}{8}=\frac{1}{2}\phantom{\rule{0ex}{0ex}}\frac{EF}{BC}=\frac{5}{10}=\frac{1}{2}\phantom{\rule{0ex}{0ex}}\frac{DF}{AC}=\frac{6}{12}=\frac{1}{2}$

So all the corresponding sides of both the triangles are equal.

$\Rightarrow \frac{DE}{AB}=\frac{EF}{BC}=\frac{DF}{AC}$

By using the SSS similarity theorem, both the triangles id="2618793" role="math" $\u25b3ABC$ and id="2618794" role="math" $\u25b3DEF$ are similar.

Find the value of x by using the SSS similarity theorem.

**Solution:**

First we find the proportion of the corresponding sides. For that, we take into account any one of the sides with unknown value. Let us consider sides AB and BC in $\u25b3ABC$ and sides DE and EF in $\u25b3DEF.$

So the proportion of the sides will be,

$\frac{AB}{DE}=\frac{BC}{EF}\phantom{\rule{0ex}{0ex}}\Rightarrow \frac{4}{12}=\frac{x-1}{18}\phantom{\rule{0ex}{0ex}}\Rightarrow 4\times 18=(x-1)\times 12\phantom{\rule{0ex}{0ex}}\Rightarrow 72=12x-12\phantom{\rule{0ex}{0ex}}\Rightarrow 12x=72+12\phantom{\rule{0ex}{0ex}}\Rightarrow 12x=84\phantom{\rule{0ex}{0ex}}\Rightarrow x=\frac{84}{12}\phantom{\rule{0ex}{0ex}}\Rightarrow x=7$

So the value of x is 7. But let us confirm it by substituting it in the unknown values sides and checking the proportions of it.

$BC=x-1=7-1=6\phantom{\rule{0ex}{0ex}}DF=3(x+1)=3(7+1)=3\times 8=24$

Now we check the proportions for the corresponding sides.

$\Rightarrow \frac{AB}{DE}=\frac{4}{12}=\frac{1}{3}\phantom{\rule{0ex}{0ex}}\frac{BC}{EF}=\frac{6}{18}=\frac{1}{3}\phantom{\rule{0ex}{0ex}}\frac{AC}{DF}=\frac{8}{24}=\frac{1}{3}$

As the given triangles are similar triangles, their proportional corresponding side should be equal. And we clearly see that they are equal. Hence our value of $x=7$ is correct.

## SSS Theorem - Key takeaways

- SSS theorem : If all the three corresponding sides of two triangles are equal to each other, then they are congruent.
- SSS congruence theorem : The two triangles are congruent if all the three respective sides of both the triangles are equal.
- SSS Similarity Theorem : Two triangles are said to be similar when the corresponding sides of these two triangles are proportional.

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##### Frequently Asked Questions about SSS Theorem

What is SSS a congruence theorem?

The two triangles are congruent if all the three respective sides of both the triangles are equal.

How do you solve SSS theorem?

SSS theorem can be solved by taking equal corresponding sides.

How do you prove SSS similarity theorem?

SSS similarity theorem is proved by using AA - similarity and SSS congruence theorem.

What is an example of SSS similarity theorem?

An example of SSS similarity theorem is one triangle with sides 9,15,18 and another triangle with sides 6,10,12.

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