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In this section, we will understand the SAS theorem and its important statements. We will also see how we can find the area of any given triangle using the SAS theorem.

## SAS congruence theorem

SAS congruence theorem gives the congruent relation between two triangles. When all the corresponding angles are congruent to each other and all the corresponding sides are congruent in two triangles, those triangles are said to be congruent. But with the SAS congruence theorem, we only consider two sides and one angle to establish the congruence between the triangles.

Here as the name suggests, SAS stands for Side-Angle-Side. When using the SAS congruence theorem, we consider two corresponding adjacent sides and the angle included between those two sides.

One should always note that the angle should be the included one and not any other, as it would not then satisfy the SAS criterion.

**SAS congruence theorem** : Two triangles are said to be congruent if the two corresponding sides and the angle included to these sides of one triangle are equal to the corresponding sides and the included angle of the other triangle.

Mathematical we represent as, if $AB=XY,\angle A=\angle X,AC=XZ,$ then $\u25b3ABC\cong \u25b3XYZ.$

If the SAS congruence theorem satisfies for any two triangles, then we can directly say that all the sides and angles of one triangle will be equal to the other triangle respectively.

## SAS similarity theorem

We can conclude two triangles are similar using the SAS similarity theorem. Usually, we need information about all the sides and angles of both the triangles to prove them similar. But with the help of the SAS similarity theorem, we only consider two corresponding sides and one corresponding angle of these triangles.

As SAS triangles have two sides, we can take the proportion of these sides to show the similarity between the two triangles.

**SAS similarity theorem** : Two triangles are similar if the two adjacent sides of one triangle are proportional to the two adjacent sides of another triangle and the included angles of both triangles are equal.

Mathematically we say that, if $\frac{AB}{DE}=\frac{BC}{EF}$ and $\angle B=\angle E,$ then $\u25b3ABC~\u25b3DEF.$

## SAS theorem proof

Let us look at the SAS theorem proof for both congruence and similarity.

### SAS congruence theorem proof

Let's perform an activity to prove the SAS congruence theorem. From the statement of the SAS congruence theorem, it is given that $AB=XY,AC=XZ,$ and $\angle A=\angle X.$

To prove: $\u25b3ABC\cong \u25b3XYZ$

We take a triangle $\u25b3XYZ$ and place it over the other triangle $\u25b3ABC.$ Now it can be seen that the B coincides with Y as $AB=XY.$ As $\angle A=\angle X$ and when both triangles are placed over each other, AC and XZ will fall alongside each other. As $AC=XZ,$ point C coincides with Z. Also BC and YZ will coincide with each other.

So clearly, both triangles $\u25b3ABC$ and $\u25b3XYZ$ coincide with each other. Hence $\u25b3ABC\cong \u25b3XYZ$.

### ASA similarity theorem proof

It is given from the statement of similarity theorem that $\frac{AB}{DE}=\frac{BC}{EF}$ and $\angle B=\angle E.$

To prove: $\u25b3ABC~\u25b3DEF$

Consider a point P on DE at such a distance such that $AB=EP.$ Then join a line segment from P to EF at point Q such that $PQ\parallel DF.$ As $PQ\parallel DF,$ we get that $\u25b3PEQ~\u25b3DEF.$

Then by Basic Proportionality Theorem,

$\frac{EP}{DE}=\frac{EQ}{EF}\left(1\right)$

**Basic Proportionality Theorem** : If a line is parallel to one side of a triangle and if that line intersects the other two sides at two different points in the triangle then those sides are in proportion.

We are already given that,

$\frac{AB}{DE}=\frac{BC}{EF}\left(2\right)$

As $AB=EP$ and from equation (1) and equation (2),

$\frac{EP}{DE}=\frac{AB}{DE}=\frac{EQ}{EF}=\frac{BC}{EF}\phantom{\rule{0ex}{0ex}}\Rightarrow EQ=BC$

And we also have that $\angle B=\angle E.$

So using the SAS congruence theorem from the above information we get that $\u25b3ABC\cong \u25b3PEQ.$

$\Rightarrow \u25b3ABC~\u25b3PEQ$

We already have that $\u25b3PEQ~\u25b3DEF.$ So from the both obtained similarity we get that $\u25b3ABC~\u25b3DEF.$

## SAS theorem formula

The SAS theorem is not only used to show congruence and similarity between two triangles, but we get the SAS theorem formula from it. This SAS formula can be very helpful in trigonometry to calculate the area of a triangle. This formula uses trigonometry rules to find the area of the triangle.

The SAS theorem formula for the triangle is expressed as :

Area of triangle$=\frac{1}{2}\times a\times b\times \mathrm{sin}x,$ where a and b are the two sides of SAS triangle and x is the measure of the included angle.

Let us derive the SAS theorem formula. In the SAS $\u25b3ABC$ construct a perpendicular from point A onto the line BC at D. Now as $\u25b3ABD$ forms a right triangle, we can use trigonometric ratios with $\angle B$ as the angle.

$\Rightarrow \mathrm{sin}x=\frac{AD}{AB}\phantom{\rule{0ex}{0ex}}\Rightarrow p=a\times \mathrm{sin}x\left(1\right)$

Also, we know that the general formula to calculate any triangle is

Area $=\frac{1}{2}\times base\times height$

Now in the $\u25b3ABC,$ b is the base and p is the height. Then substituting this value in the formula of are we get,

Area $=\frac{1}{2}\times b\times p\left(2\right)$

From equation (1) we know the value of p, so we substitute that in the above-obtained equation (2) of area.

Area $=\frac{1}{2}\times b\times a\times \mathrm{sin}x$

Hence the formula for the area of the triangle using the SAS theorem is $\mathit{A}\mathit{r}\mathit{e}\mathit{a}\mathbf{=}\frac{\mathbf{1}}{\mathbf{2}}\mathbf{\times}\mathit{a}\mathbf{\times}\mathit{b}\mathbf{\times}\mathit{s}\mathit{i}\mathit{n}\mathbf{}\mathit{x}.$

## SAS theorem examples

Here are some of the SAS theorem examples to understand the concept better.

Determine if the given triangles are similar or not.

**Solution:**

From the figure, we can see that two sides and one angle measure are provided for each triangle. And the gives are adjacent and the angle is the included angle of both the sides. so the given triangle can be considered as the SAS triangle.

Here we need to determine the similarity between $\u25b3ABC$ and $\u25b3XYZ.$ But for that, we need to verify the SAS similarity theorem.

As $\angle B$ and $\angle Z$ are both right-angle triangles. So it implies that $\angle B\cong \angle Z.$

We also need to check the proportion between the given sides.

$\Rightarrow \frac{XZ}{AB}=\frac{10}{15}=\frac{2}{3},\phantom{\rule{0ex}{0ex}}\frac{YZ}{BC}=\frac{24}{36}=\frac{2}{3}\phantom{\rule{0ex}{0ex}}\Rightarrow \frac{AB}{XZ}=\frac{BC}{YZ}$

So, from above we can see that both sides and an angle satisfy the condition of the SAS similarity theorem. Hence both the triangles $\u25b3ABC$ and $\u25b3XYZ$ are similar triangles.

Find the area of the given triangle$\u25b3DEF$ using SAS theorem formula, if $EF=12cm,DF=10cm,$ and $\angle F=30\xb0.$

**Solution:**

Here we are given that $EF=12cm,DF=10cm,\angle F=30\xb0.$ So consider $a=10cm,b=12cm,x=30\xb0.$

Then Area of the above triangle using the SAS theorem formula is,

Area $=\frac{1}{2}\times a\times b\times \mathrm{sin}x$

$=\frac{1}{2}\times 10\times 12\times \mathrm{sin}30\xb0\phantom{\rule{0ex}{0ex}}=\frac{1}{2}\times 10\times 12\times \frac{1}{2}\phantom{\rule{0ex}{0ex}}=5\times 6$

$\therefore \mathit{A}\mathit{r}\mathit{e}\mathit{a}\mathbf{=}\mathbf{30}\mathbf{}\mathit{c}{\mathit{m}}^{\mathbf{2}}$

Hence, the area of the triangle using the SAS theorem formula is 30 cm^{2}

## SAS Theorem - Key takeaways

- SAS congruence theorem : Two triangles are said to be congruent if the two corresponding sides and the angle included to these sides of one triangle are equal to the corresponding sides and the included angle of the other triangle.
- SAS similarity theorem : Two triangles are similar if the two adjacent sides of one triangle are proportional to the two adjacent sides of another triangle and the included angles of both triangles are equal.
- Area of a triangle using SAS theorem $=\frac{1}{2}\times a\times b\times \mathrm{sin}x,$ where a and b are the two sides of SAS triangle and x is the measure of the included angle.

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##### Frequently Asked Questions about SAS Theorem

What is SAS theorem?

SAS Theorem gives the congruence and similarity relation of two triangles with corresponding sides and included angle of both the triangles.

How do you Prove the SAS Congruence Rule?

SAS congruence rule can be proved by superimposing both the triangles on each other.

How do you solve the SAS similarity theorem?

SAS similarity triangle can be solved using the basic proportionality theorem.

What is the SAS Theorem example?

Suppose for triangles ABC and XYZ, if AB=XY =5, BC=YZ=10, and the included angle between both the triangle is 45°.

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