#### CHAPTER AT A GLANCE

##### Similar Figures

Two figures having the same shape but not necessarily the same size are called similar figures/

Two figures having the same shape as well as same size are called congruent figures.

**Note :** that all congruent figures are similar but the similar figures need not be congruent.

##### Similarity of Triangles

Two triangles are similar, if

(i) their corresponding angles are equal and

(ii) their corresponding sides are in the same ratio (or proportion)

**NOTE :** If corresponding angles of two triangles are equal, then they are known as

##### Equiangular Triangles.

**Criteria for Similarity of Triangles**

**(i) AAA Similarity Criterion :** If in two triangles, corresponding angles are equal, then their corresponding sides are in the same ratio and hence the two triangles are similar.

**(ii) AA Similarity Criterion :** If in two triangles, two angles of one triangle are respectively equal to the two angles of the other triangle, then the two triangles are similar.

**(iii) SSS Similarity Criterion : **If in two triangles, corresponding sides are in the same ratio then their correspoding angles are equal and hence the triangles are similar.

**(iv) SAS Similarity Criterion :** If one angle of a triangle is equal to one angle of another triangle and the sides including these angles are in the same ratio (proportion), then the two triangles are similar.

##### Theorems and some Important Results

**Theorem : **Basic Proportionality Theorem or Thale’s Theorem**Statement:** “If a line is drawn parallel to one side of a triangle, it divides the other two sides in the same ratio”

In ABC, a line parallel to BC intersects AB at D and AC at E. then

AD/DB = AE/EC

**Theorem: Converse of B.P. Theorem**

**Statement: “If a line divides any two sides of a triangle in the same ratio,”**

##### Areas of Two Similar Triangles

**Theorem 1: Prove that the ratio of the areas of two similar triangles is equal to the ratio of the squares of their corresponding sides**.

**Theorem 2: Prove that the areas of two similar triangles are in the ratio of the square of corresponding altitudes.**

**Theorem: Pythagoras Theorem or Bodhayan Theorem**

**Statement: Prove that,”In a right angled triangle, the square of the hypotenuse is equal to the sum of the other two sides”.**

**Theorem: **Converse of Pythagoras Theorem

**Statement: **“In a triangle, the square of the longer side is equal to the sum of the squares of the other two sides then the triangle is right angled triangle”.

ABC is a triangle such that BC2 = AB2 + AC2 then LA = 90Â°

**Related Articles:**