Inequality relations in a triangle:
(1) If two sides of a triangle are unequal, the greater side has the greater angle opposite to it.
(2) If two angles of a trangle are unequal, the greater angle has the greater side opposite to it.
(3) Any two sides of a triangle .are together greater than the third side.
(4) The difference of any two sides of a triangle is less than the third side. (5) The sum of any two sides of a triangle is greater than the third side.
(1) Similarity of two figures:
Figures which are of the same shape but not necessarily of the same size are called similar figures. The symbol stands for 'is similar to'.
Two polygons are said to be similar to each other if:
(i) Their corresponding angles are equal, and
(ii) The lengths of their corresponding sides are proportional.
Similar triangles: Two triangles are said to be similar if:
(i) Their corresponding angles are equal, and
(ii) Their corresponding sides are proportional.
(2) Condition for similarity of two triangles or three similarity postulates for triangles:
Postulate1: (SAS) Similarity: If a pair of corresponding angles are equal and the sides including them are proportional, then the triangles are similar.
Postulate 2: (AA) Similarity: If two pairs of corresponding angles are equal, then triangles are similar.
Postulate 3: (SSS) Similarity: If three pairs of corresponding sides are proportional then the two triangles are similar.
(3) Characteristic Properties of
Similar Triangles:
(i) AAA Similarity or AA Similarity.
(ii) SSS Similarity.
(iii) SAS Similarity.
(4) Similar triangles are not necessarily congruent: To establish the similarity of two Ds, it is sufficient to satisfy one condition. The Ds are similar if
(i) Their corresponding angles are equal.
(ii) Their corresponding sides are proportional.
Note: If the corresponding angles are equal, the Ds are equiangular.
Basic proportionality theorem: If a line is drawn parallel to one side of a triangle intersecting the other two sides are divided in the same ratio.
Adding 1 to both sides we get 
A perpendicular drawn from the right angle vertex of a right-angled triangle divides the triangle into two triangles similar to each other and also to the original triangle.
Then ΔBCO ≈ ΔABO ≈ ΔABC.
Note: Congruent triangles are equal in all respects their angles are equal, their sides are equal and their areas are equal. Congruent triangles are always similar.
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