Math's: Similarity of Triangles: Similar Figures, Polygons and Triangles

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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 Figures

Similar Figures

Similar Polygons

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

Similar Polygons

Similar Polygons

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

Similar Triangles

Similar Triangles

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.

Similar Triangles

Similar Triangles

  • 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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