What is Similarity of Triangles?

Two shapes are called similar if they have the same shape but may have different sizes.

For triangles, similarity means:

• All corresponding angles are equal.
• All corresponding sides are in the same ratio.

 △ABC  ∼  △DEF
 Same shape, size may differ.

Definition: Two triangles are said to be similar if their corresponding angles are equal and their corresponding sides are in proportion.

Symbolically, if:

• \( \angle A = \angle D \)
• \( \angle B = \angle E \)
• \( \angle C = \angle F \)

and

\( \dfrac{AB}{DE} = \dfrac{BC}{EF} = \dfrac{CA}{FD} \)

then the triangles are similar:

\( \triangle ABC \sim \triangle DEF \)

• Corresponding angles are equal.
• Corresponding sides are in the same ratio.
• The shape is identical, but size may differ.
• Areas of similar triangles are in the ratio of the squares of their corresponding sides.

Similarity is very helpful in geometry because it allows us to find unknown lengths without constructing triangles physically. It is used in:

• Maps and scale drawings,
• Shadow and height problems,
• Trigonometry,
• Surveying and architecture.

Triangles can be proven similar using these basic criteria:

• AAA Similarity
• SAS Similarity
• SSS Similarity

Each criterion will be covered in separate topics.

      A                D
     / \              / \
    /   \            /   \
   B-----C          E-----F
Same shape, different sizes → Similar