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Similarity in Geometry

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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.

Similarity

  • Any two circles are similar.

Similarity in Geometry

  • Two squares are always similar.
Similarity Definition
  • All the cubes are similar.

Similar Cubes

Any two angles with the same measure are similar irrespective to length of their arms.
Similar Angles
Thus, any two figures with the same shape are said to be similar shapes.

Similarity Definition

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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.
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 $\bigtriangleup DEF$.
Similar Triangles
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.
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Similar Polygons

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Polygons are the closed curves formed by joining three or more line segments. Two polygons are similar, if they have same shape, but not necessarily the same size. All similar polygons have equal measure of their corresponding angles and thus, they form the same shape. Following figures demonstrates similar polygons:
Figure a:
Similar Polygons
Figure b:
Similar Polygon
Here, all the angles of one polygon are equal to the corresponding angles of another polygon and ratio of all corresponding sides are the same.
For example, in figure (a), we have:
$\angle A=\angle P$, $\angle B=\angle Q$, $\angle C=\angle R$, $\angle D=\angle S$, $\angle E=\angle T$

$\frac{AB}{PQ}=\frac{BC}{QR}=\frac{CD}{RS}=\frac{DE}{ST}=\frac{AE}{PT}$.