Let’s discuss each topic of this chapter one-by-one

Basic Proportionality Theorem / Thales Theorem

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. Let’s solve a question based on this theorem:

Similar triangles & Similarity criterion

We know that 2 triangles are said to be similar when their shape is exactly same, i.e.,

• Their corresponding angles are equal
• Their corresponding sides are in the same ratio (or proportion)

Their size can vary but not shape. Remember that all the congruent figures are similar but the converse is not true. The similarity criterion are some conditions that we look in triangles which are sufficient to prove that triangles are congruent. These criteria are:

• AAA (Angle – Angle – Angle)- If in two triangles, corresponding angles are equal, then their corresponding sides are in the same ratio (or proportion) and hence the two triangles are similar.
• AA (Angle – Angle) – If two angles of one triangle are respectively equal to two angles of another triangle, then the two triangles are similar.
• SSS (Side – Side – Side): If in two triangles, sides of one triangle are proportional to (i.e., in the same ratio of ) the sides of the other triangle, then their corresponding angles are equal and hence the two triangles are similar.
• SAS (Side – Angle – Side): If one angle of a triangle is equal to one angle of the other triangle and the sides including these angles are proportional, then the two triangles are similar.

In the questions generally asked in boards, we do not have to just prove 2 triangles similar but use the result to either find some value or prove statements. For example:

Ratio of area of similar triangles

For similar triangles, the ratio of their areas is equal to the square of the ratio of their corresponding sides, i.e.,

Pythagoras Theorem & its converse

We have all studied Pythagoras Theorem in previous classes and it states that: “In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides”.

Its converse theorem states that: “If in a triangle, 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”.

Let’s solve one of the questions: