congruent polygons — Definition, Formula & Examples
Congruent polygons are polygons that have exactly the same shape and size, meaning you could place one directly on top of the other and they would match perfectly.
Two polygons are congruent if and only if there exists a one-to-one correspondence between their vertices such that all corresponding sides are equal in length and all corresponding angles are equal in measure.
How It Works
To show two polygons are congruent, you need to match each vertex of one polygon to a vertex of the other so that every pair of corresponding sides has the same length and every pair of corresponding angles has the same measure. The order in which you list the vertices matters — it defines which sides and angles correspond. For triangles, shortcut tests like SSS, SAS, and ASA let you prove congruence without checking every side and angle. For polygons with four or more sides, no such shortcut exists; you generally must verify all corresponding sides and all corresponding angles match.
Worked Example
Problem: Quadrilateral ABCD has sides AB = 5, BC = 8, CD = 5, DA = 8 and angles A = 70°, B = 110°, C = 70°, D = 110°. Quadrilateral PQRS has sides PQ = 5, QR = 8, RS = 5, SP = 8 and angles P = 70°, Q = 110°, R = 70°, S = 110°. Are the two quadrilaterals congruent?
Step 1: Set up the correspondence A↔P, B↔Q, C↔R, D↔S and compare corresponding sides.
Step 2: Compare corresponding angles under the same correspondence.
Step 3: Since all corresponding sides and all corresponding angles are equal, the polygons are congruent.
Answer: Yes, quadrilateral ABCD is congruent to quadrilateral PQRS.
Why It Matters
Polygon congruence is central to geometric proofs in high-school geometry courses. Engineers and architects rely on congruent shapes when designing parts that must be identical, such as floor tiles or machine components that need to fit interchangeably.
Common Mistakes
Mistake: Listing vertices in the wrong order and then comparing non-corresponding sides or angles.
Correction: The vertex order in the congruence statement defines which parts correspond. If you write ABCD ≅ PQRS, then A↔P, B↔Q, C↔R, D↔S. Always check that the correspondence makes all pairs match.