Similarity of Triangles

In geometry, triangles are similar when they share the same shape, irrespective of their size. This happens when:

1. Corresponding angles are equal.
2. Corresponding sides are in the same ratio (proportion).

For example, if , then:

This concept is foundational in geometry and has practical applications in fields such as construction, navigation, and design.

Basic Proportionality Theorem (Thales' Theorem)

Statement

If a line is drawn parallel to one side of a triangle, intersecting the other two sides, the line divides those sides in the same ratio.

In , if , then:

Understanding Thales' Theorem

Setup

1. Draw any angle and on one arm , mark points such that:
2. Draw another arm and mark a point on it.
3. From , draw .

Observations

Measure , , , and .
You will find:

This supports the Basic Proportionality Theorem.

Proof of Basic Proportionality Theorem

Given

In , intersects and at and .

To Prove

Construction

1. Draw and .

Proof

1. Area of is proportional to its base :
   
2. Similarly, for :
   
3. Taking the ratio of areas:
   Area of
4. Similarly, for and :
   Area of
5. Thus, .

Converse of Basic Proportionality Theorem

Statement

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

Proof

1. Suppose .
2. Assume is not parallel to .
3. Draw , and by the Basic Proportionality Theorem:
4. But this contradicts . Hence, .

Examples Based on Theorems

Example 1: Prove

Solution:
From Thales' Theorem:

​

Adding 1 to both sides:

​

Thus:

Example 2: Prove in a trapezium.

Solution:
Join , intersecting at .

1. Using Thales' Theorem in :
2. Similarly, in :

Thus:

Example 3: Prove .

Solution:

1. , so by Thales' Theorem:
2. (corresponding angles).

Thus, .

Practice Problems

1. In , prove that when .
2. Prove that the ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides.
3. In , . If cm, cm, and cm, find .