**Important Concepts and Formulas**

1. Two figures having the same shape but
not necessarily 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 (that is proportion).

4.

**Theorem 1:**If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, then the other two sides are divided in the same ratio.
5.

**Theorem 2:**If a line divides any two sides of a triangle in the same ratio, then the line is parallel to the third side.
6.

**Theorem 3:**If in two triangles, corresponding angles are equal, then their corresponding sides are in the same ratio and hence the two triangles are similar. This is called AAA similarity criterion.
7.

**Theorem 4:**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. This is called AA similarity criterion.
8.

**Theorem 5:**If in two triangles, corresponding sides are in the same ratio, then their corresponding angles are equal and hence the triangles are similar. This is called SSS similarity criterion.
9.

**Theorem 6:**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, then the triangles are similar. This is called SAS similarity criterion.
10.

**Theorem 7:**The ratio of the areas of two similar triangles is equal to the ratio of their corresponding sides.
11.

**Theorem 8:**If a perpendicular is drawn from the vertex of the right angle of a right-angled triangle to the hypotenuse, then the triangles on both sides of the perpendicular are similar to the whole triangle and also to each other.
12.

**Theorem 9:**In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. This theorem is called**Pythagoras theorem**. It states that (Hypotenuse)^{2}= (Base)^{2}+ (Height)^{2}^{}To study Pythagoras theorem in detail -------- Click Here!
13.

**Theorem 10:**If in a triangle, the square of one side is equal to the sum of the squares of the other two sides, then angle opposite the first side is a right angle and the triangle is the right-angled triangle. This is called the**converse of the Pythagoras theorem**.