SSS Theorem

SSS congruence theorem

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

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

SSS congruence triangles, Mouli Javia - StudySmarter Originals

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 $△ABC$ and $△XYZ$ 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 $△ABC\cong △XYZ.$

SSS congruence triangle examples

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

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.

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: $△ABC~△MNO$

Triangles with constructed parallel line, Mouli Javia - StudySmarter Originals

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.

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 $△MPQ~△MNO.$

From the definition of similar triangles on $△MPQ$ and $△MNO,$ 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" $△ABC\cong △MPQ.$And we also have that id="2618785" role="math" $△MPQ~△MNO.$ Hence from both the similarity we get id="2618788" role="math" $△ABC~△MNO.$

SSS similarity theorem examples

Let us take a look at SSS similarity theorem examples.