# Math՚s: Similarity of Triangles: Similar Figures, Polygons and Triangles (For CBSE, ICSE, IAS, NET, NRA 2022)

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## Similar Figures

Figures which have the same shape but not necessarily the same size are called similar figures.

• Two line-segments of the same length are congruent as well as similar and of different lengths are similar but not congruent.
• Two circles of the same radius are congruent as well as similar and circles of different radii are similar but not congruent.
• Two equilateral triangles of different sides are similar but not congruent.
• Two squares of different sides are similar but not congruent.

## Similar Polygons

Any two polygons, with corresponding angles equal and corresponding sides proportional, are similar.

In the above two pentagons, . Therefore, these pentagons are similar.

In case of a rectangle and a square, although the angles are equal (all are right angles) , but the corresponding sides are not proportional. So, a rectangle and a square are not similar figures.

## Similar Triangles

Triangles are special type of polygons and therefore the conditions of similarity of polygons also hold for triangles. Thus, two triangles are similar if-

• their corresponding angles are equal, and
• their corresponding sides are proportional.
• In and , and
• Thus, we say that is similar to and denote it as

## Criteria for Similarity of Triangles

We have three important criteria for the similarity of triangles-

• AAA (Angle- Angle- Angle) Criterion- If in two triangles, all the three corresponding angles are equal, the triangles are similar.
• SSS (Side- Side- Side) Criterion- If the three corresponding sides of two triangles are proportional the triangles are similar.
• SAS (Side- Angle- Side) Criterion- If one angle of a triangle is equal to one angle of the other triangle and the sides containing these angles are proportional, the triangles are similar.

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