Similarity of Triangles
* Two triangles are similar, if their corresponding angles are equal and their corresponding sides are proportional or in the same ratio.
* Basic proportionality Theorems: If a line is drawn parallel to one side of triangle to intersect the other two sides in distinct points, the other two sides are divided in the same ratio.
* Converse of the Proportionality Theorem: if a line divides any two sides of a triangle in the same ratio, then the line is parallel to the third side.
Lines parallel to the same line are parallel to each other.
Criteria for Similarity of Triangles
* Two triangles are similar, if their corresponding angles are equal and their corresponding sides are in same ratio (or proportion).
* If in two triangles, the corresponding angles are equal, then their corresponding sides are in the same ratio (or proportion) and the angles are similar. This criterion is referred to as the AAA (Angle-Angle-Angle) criterion of similarity of two triangles.
* If two angles of one triangle are respectively equal to two angles of another triangle, then the two triangles are similar. This is referred to as AA similarity
criterion.
* If in two triangles, the sides of one triangle are proportional to (i.e., in the same ratio of) the side of the other triangle, then their corresponding angles are equal and the triangles are similar. This criterion is referred to as the SSS (Side-Side-Side) similarity criterion for two triangles.
* If one angle of the triangle is equal to one angle of the other triangle and the sides including these angles are proportional, then the triangles are similar. This criterion is referred to as the SAS (side-angle-side) similarity criterion for two triangles.
Areas of Similar Triangles
* The ratio of areas of two similar triangles is equal to the square of the ratio of their corresponding sides.
* If perpendicular is drawn from the vertex of the right angle of a right triangle to the hypotenuse, then triangles on the both sides of perpendicular are similar to the whole triangle and to each other.
* In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
* In a triangle, if square of one side is equal to the sum of the squares of the other two sides, then the angle opposite the first side is a right angle.