Congruence Formula

The Formula

\triangle ABC \cong \triangle DEF \Leftrightarrow all corresponding sides and angles are equal

When to use: If you could pick up one shape and place it exactly on the other, they're congruent.

Quick Example

Two triangles with sides 3-4-5 are congruent—every side and angle matches exactly.

Notation

\cong means 'is congruent to'

What This Formula Means

Two geometric figures are congruent if they have exactly the same size and shape, so one can be placed on the other perfectly.

If you could pick up one shape and place it exactly on the other, they're congruent.

Formal View

F_1 \cong F_2 \iff \exists isometry T: \mathbb{R}^2 \to \mathbb{R}^2 such that T(F_1) = F_2; for triangles: \triangle ABC \cong \triangle DEF \iff |AB|=|DE|, |BC|=|EF|, |AC|=|DF| and \angle A=\angle D, \angle B=\angle E, \angle C=\angle F

Worked Examples

Example 1

easy
Triangle ABC has sides 3 cm, 4 cm, 5 cm. Triangle DEF has sides 3 cm, 4 cm, 5 cm. Are they congruent?

Solution

  1. 1
    Step 1: Congruent figures have exactly the same size and shape.
  2. 2
    Step 2: Compare corresponding sides: AB=DE=3, BC=EF=4, AC=DF=5.
  3. 3
    Step 3: All three pairs of sides are equal, so by SSS (Side-Side-Side) congruence, the triangles are congruent.
  4. 4
    Step 4: Write the congruence statement: \triangle ABC \cong \triangle DEF.

Answer

Yes, \triangle ABC \cong \triangle DEF by SSS.
SSS congruence states that if all three sides of one triangle equal all three sides of another, the triangles must be identical in shape and size. The angles are automatically determined by the side lengths.

Example 2

medium
Two rectangles: Rectangle 1 has dimensions 4 cm × 6 cm. Rectangle 2 has dimensions 6 cm × 4 cm. Are they congruent? Explain.

Common Mistakes

  • Confusing with similarity
  • Not checking all corresponding parts

Why This Formula Matters

Foundation for proofs and understanding equal geometric objects.

Frequently Asked Questions

What is the Congruence formula?

Two geometric figures are congruent if they have exactly the same size and shape, so one can be placed on the other perfectly.

How do you use the Congruence formula?

If you could pick up one shape and place it exactly on the other, they're congruent.

What do the symbols mean in the Congruence formula?

\cong means 'is congruent to'

Why is the Congruence formula important in Math?

Foundation for proofs and understanding equal geometric objects.

What do students get wrong about Congruence?

Students think congruent means 'same shape' but forget it also means 'same size.' Two shapes can look similar without being congruent.

What should I learn before the Congruence formula?

Before studying the Congruence formula, you should understand: shapes, equal.

Want the Full Guide?

This formula is covered in depth in our complete guide:

Symmetry, Rotational Symmetry, and Congruence →
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