**1.** Two figures having the same shape but not neces-sarily the same size are called similar figures.

**2.** All the congruent figures are similar but the converse is not true.

**3.** Two polygons of the same number of sides are similar, if (i) their corresponding angles are equal and (ii) their corresponding sides are in the same ratio (i.e., proportion).

**4.** Two triangles are similar, if

(i) their corresponding angles are equal

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

**5. Basic Proportionality Theorem (B.P.T.) (Thales Theorem)**

In a triangle, a line drawn parallel to one side, to intersect the other sides in distinct points, divides the two sides in the same ratio.

In , if then (i)

(ii) (iii)

**6. Converse of Basic A Proportionality Theorem**

If a line divides any two sides of a triangle in the same ratio, the line is parallel to the third side.

In , if , then .

**7.** If in two triangles, corresponding angles are equal, then their corresponding sides are in the same ratio and hence the two triangles are similar (AAA similarity criterion).

**8.** 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 (AA similarity criterion).

**9.** If in two triangles, corresponding sides are in the same ratio, then their corresponding angles are equal and hence the triangles are similar (SSS similarity criterion).

**10.** If one angle of a triangle is equal to the one angle of another triangle and the sides including these angles are in the same ratio (proportional), then the triangles are similar (SAS similarity criterion).

**11.** If a perpendicular is drawn from the vertex of the right angle of a right triangle to hypotenuse, then the triangles on both sides of the perpendicular are similar to the whole triangle and also to each other.

**12.** The ratio of the areas of two similar triangles are equal to the ratio of the squares of any two corresponding sides.

**13.** The areas of two similar triangles are in the ratio of the squares of the corresponding altitudes.

**14.** The areas of two similar triangles are in the ratio of the squares of the corresponding medians.

**15.** If the areas of two similar triangles are equal, then the triangles are congruent, i.e., equal and similar triangles are congruent.

**16. The Pythagoras Theorem**

In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. In figure, , so

**17. Converse of the Pythagoras Theorem**

In a triangle, if the square of one side is equal to the sum of the squares of the other two sides, then the angle opposite to the first side is a right angle.

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