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# Similarity in Geometry

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 Sub Topics The two figures having same shape are said to be geometrically similar. Few examples of similarity are shown below:Two straight lines are similar to each other. Any two circles are similar. Two squares are always similar. All the cubes are similar. Any two angles with the same measure are similar irrespective to length of their arms.Thus, any two figures with the same shape are said to be similar shapes.

## Similarity Definition

We can define similarity as follows: "Two geometrical figures are known as similar figures, if either both the objects have same shape or one possesses same shape as the mirror image of another". We can say that if two images are similar, then by shrinking or enlarging one image, other can be obtained.

The difference between similarity and congruency is that similar figures have same shape, whereas congruent figures have same shape and size.

For similar figures, size does not matter. This means that two similar figures not necessarily are of same size. If the dimensions of two geometrical figures are different, but their shapes are exactly same, then the figures are said to be similar figures.

## Similar Triangles

Two triangles are similar if they are equiangular. For two similar triangles:
• All corresponding angles are equal.
• All corresponding sides are proportional.
Let us consider two similar triangles $\bigtriangleup ABC$ and $\bigtriangleup DEF$.

Then, by the definition of similar triangles, we have:
$\angle A=\angle D$

$\angle B=\angle E$

$\angle C=\angle F$

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

We denote similarity by the symbol "$\sim$". So, for the above two similar triangles, we write:
$\bigtriangleup ABC\sim \bigtriangleup DEF$

Mathematically, two triangles are said to be similar, if one of the following three criterias hold:
• AAA or AA criterion: Two triangles are similar if either all the three corresponding angles are equal or any two corresponding angles are equal. AAA and AA criteria are same because if two corresponding angles of two triangles are equal, then third corresponding angle will definitely be equal.
• SSS criterion: Two triangles are said to be similar, if all the corresponding sides are in the same proportion.
• SAS criterion: Two triangles are similar if their two corresponding sides are in the same proportion and the corresponding angles between these sides are equal.
If any one of the above criteria is satisfied, then the triangles are known as similar triangles and hence their corresponding sides will be in same ratio.

## Similar Polygons

$\angle A=\angle P$, $\angle B=\angle Q$, $\angle C=\angle R$, $\angle D=\angle S$, $\angle E=\angle T$
$\frac{AB}{PQ}=\frac{BC}{QR}=\frac{CD}{RS}=\frac{DE}{ST}=\frac{AE}{PT}$.