Congruence Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Congruence.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

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.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Congruence preserves all measurements—it's about exact sameness.

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

Sense of Study hint: Try cutting out both shapes and placing one on top of the other. If they match exactly with no gaps, they are congruent.

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.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
Are a square with side 5 cm and a rhombus with side 5 cm necessarily congruent? Explain why or why not.

Example 2

hard
Triangle PQR: \angle P = 50°, \angle Q = 60°, PQ = 8 cm. Triangle XYZ: \angle X = 50°, \angle Y = 60°, XY = 8 cm. Are they congruent? Which postulate applies?
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Background Knowledge

These ideas may be useful before you work through the harder examples.

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