Master the Similarity of Triangles: Detailed Concepts, Proofs, and Theorems

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

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

For example, if $\boldsymbol$ , then:

$\boldsymbol$ $\displaystyle\boldsymbol$

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 $\boldsymbol$ , if $\boldsymbol$ , then:

$\displaystyle\boldsymbol$

Understanding Thales' Theorem

Setup

Draw any angle $\boldsymbol$ and on one arm $\boldsymbol$ , mark points $\boldsymbol$ such that: $\boldsymbol$
Draw another arm $\boldsymbol$ and mark a point $\boldsymbol$ on it.
From $\boldsymbol$ , draw $\boldsymbol$ .

Observations

Measure $\boldsymbol$ , $\boldsymbol$ , $\boldsymbol$ , and $\boldsymbol$ .
You will find:

$\displaystyle\boldsymbol{\frac = \frac}$

This supports the Basic Proportionality Theorem.

Proof of Basic Proportionality Theorem

Given

In $\boldsymbol$ , $\boldsymbol$ intersects $\boldsymbol$ and $\boldsymbol$ at $\boldsymbol$ and $\boldsymbol$ .

To Prove

$\displaystyle\boldsymbol{\frac = \frac}$

Construction

Draw $\boldsymbol$ and $\boldsymbol$ .

Proof

Area of $\boldsymbol$ is proportional to its base $\boldsymbol$ :
$\displaystyle\boldsymbol$
Similarly, for $\boldsymbol$ :
$\displaystyle\boldsymbol$
Taking the ratio of areas:
Area of $\displaystyle\boldsymbol{\frac{\text{Area of } \triangle ADE}{\text{Area of } \triangle BDE} = \frac}$
Similarly, for $\boldsymbol$ and $\boldsymbol$ :
Area of $\displaystyle\boldsymbol{\frac{\text{Area of } \triangle ADE}{\text{Area of } \triangle DEC} = \frac}$
Thus, $\displaystyle\boldsymbol{\frac = \frac}$ .

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

Suppose $\displaystyle\boldsymbol{\frac = \frac}$ .
Assume $\boldsymbol$ is not parallel to $\boldsymbol$ .
Draw $\boldsymbol \parallel BC}$ , and by the Basic Proportionality Theorem: $\displaystyle\boldsymbol{\frac = \frac{AE'}{E'C}}$
But this contradicts $\displaystyle\boldsymbol{\frac = \frac}$ . Hence, $\boldsymbol$ .

Examples Based on Theorems

Example 1: Prove $\displaystyle\boldsymbol{\frac = \frac = \frac}$

Solution:
From Thales' Theorem:

$\displaystyle\boldsymbol$

Adding 1 to both sides:

$\displaystyle\boldsymbol$ $\displaystyle\boldsymbol$

Thus:

$\displaystyle\boldsymbol$

Example 2: Prove $\displaystyle\boldsymbol$ in a trapezium.

Solution:
Join $\boldsymbol$ , intersecting $\boldsymbol$ at $\boldsymbol$ .

Using Thales' Theorem in $\boldsymbol$ : $\displaystyle\boldsymbol$
Similarly, in $\boldsymbol$ : $\displaystyle\boldsymbol$

Thus:

$\displaystyle\boldsymbol$

Example 3: Prove $\boldsymbol$ .

Solution:

$\boldsymbol$ , so by Thales' Theorem: $\displaystyle\boldsymbol$
$\boldsymbol$ (corresponding angles).

Thus, $\boldsymbol$ .

Practice Problems

In $\boldsymbol$ , prove that $\displaystyle\boldsymbol$ when $\boldsymbol$ .
Prove that the ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides.
In $\boldsymbol$ , $\boldsymbol$ . If $\boldsymbol$ cm, $\boldsymbol$ cm, and $\boldsymbol$ cm, find $\boldsymbol$ .