What is Congruence of Triangles?

The word congruent means exactly the same in shape and size. When two figures are congruent, one can be placed on top of the other so that they match perfectly.

For triangles, congruence means all corresponding sides and all corresponding angles are equal.

   △ABC  ≅  △DEF
  (same shape, same size)

Definition: Two triangles are said to be congruent if all their corresponding sides and angles are equal.

Symbolically, if \( \triangle ABC \) is congruent to \( \triangle DEF \), we write:

\( \triangle ABC \cong \triangle DEF \)

This means:

• \( AB = DE \)
• \( BC = EF \)
• \( CA = FD \)
• \( \angle A = \angle D \)
• \( \angle B = \angle E \)
• \( \angle C = \angle F \)

Once two triangles are proven congruent, their matching parts become equal automatically. This is known as:

CPCTC – Corresponding Parts of Congruent Triangles are Congruent.

This idea is used to prove statements about sides or angles using triangle congruence.

Congruence helps us:

• prove equality of sides and angles,
• show two shapes are identical in size,
• solve geometric proofs,
• understand rigid motions (flips, slides, turns).

It forms the base for more advanced geometry topics involving properties and constructions.

To avoid measuring all sides and angles, we use special guidelines called congruence rules.

The commonly used congruence criteria are:

• SSS (Side-Side-Side)
• SAS (Side-Angle-Side)
• ASA (Angle-Side-Angle)
• AAS (Angle-Angle-Side)
• RHS (Right angle-Hypotenuse-Side)

Each of these rules will be covered in separate topics.

You see congruent triangles in many real-world situations:

• Two identical roof truss pieces,
• Repeated triangular patterns in designs or tiling,
• Matching triangular components in machines,
• Architectural detailing where identical shapes are needed.

Congruence ensures uniformity in shapes and measurements.