Congruent Polygons are polygons that have the same shape and size. When two polygons are congruent, one of the polygons can be placed exactly over the other by sliding, flipping or turning.

Congruence shapes find application widely in machine designing and manufacturing.  Equilateral triangles , squares and regular hexagons are congruent regular polygons which tessellate around a vertex to form patterns. Congruent Regular Polygons also make the faces of regular polyhedra which are called Platonic solids. The commonly known Platonic solids are Tetrahedron and Hexahedron. A Tetrahedron is made up of four congruent equilateral triangles and six congruent squares constitute a Hexahedron.

Congruent Polygons Definition

Two polygons are congruent if the vertices of one polygon bears a one to one correspondence to the regular Pole vertices of the other polygon and the corresponding angles and corresponding sides so formed are congruent. In simple words "Corresponding parts of congruent polygons are congruent."

Congruent polygons are named by placing the corresponding vertices in the same order for both the polygons.

Given that, ABCDE ≅ FGHIJ     then the corresponding vertices are
A ↔ F
B ↔ G
C ↔ H
D ↔ I
E ↔ J
The angles named after corresponding vertices are the corresponding angles and they are congruent.
∠A ≅ ∠F,   ∠B ≅ ∠G,    ∠C  ≅ ∠H,     ∠D ≅ ∠I    and ∠E ≅ ∠J.

The congruent corresponding sides are named combining corresponding vertices in the same order.
AB ≅  FGBCGHCDHIDEIJ   and EAJF.

Are Congruent Polygons Similar?

Similar polygons are polygons withe same shape, but need not be of the same size. Since congruence maintains the shape, the Congruent polygons are also similar polygons.
But conversely, the similar polygons may not be congruent polygons as similarity does not include same size in its definition.

Solving Congruent Polygons

Solved Examples

Question 1: Given Quadrilateral ABCD ≅ Quadrilateral PQRS. The angle measures and the lengths of two sides of quadrilaterals are marked on the diagram. Find the measures of the angles of quadrilateral PQRS and the lengths of sides corresponding to the sides of ABCD whose lengths are given.

Solution:

The Corresponding angles can be found by mapping the corresponding vertices in the order the quadrilaterals are named.
ABCD ≅ PQRS
∠A ≅ ∠P   ⇒ m ∠P = m ∠A = 116º.
∠B ≅ ∠Q  ⇒ m ∠Q = m ∠B = 83º.
∠C ≅ ∠R  ⇒ m ∠R = m ∠C = 83º.
∠D ≅ ∠S  ⇒ m ∠S = m ∠D = 78º.

The lengths of sides AB and BC are given.  The corresponding sides are PQ and QR.
PQ ≅  AB  ⇒  PQ = AB = 4".
QRBC  ⇒  QR = BC = 5.5".

Question 2: In the diagram given below, Δ ABC ≅ Δ DEF. Find the length EF.

Solution:

The lengths of legs AB and CA of Δ ABC are given. Using the congruence statement , the corresponding congruent
sides are,
AB DE,   BCEF  and CAFD.
BC is side corresponding to EF.  The length of BC can be found applying Pythagorean theorem,
BC2 = AB2 + CA2.
= 42 + 72 = 16 + 49 = 65
BC = √65 cm
Thus length EF = length BC = √65 cm ≈ 8.06 cm.

Question 3: A tetrahedron is a solid made up of four congruent equilateral triangles. If the measure of one of the edges = 5 cm, find the surface area of the tetrahedron nearest to the 10the of a square cm.

Solution:

The tetrahedron has four congruent triangular faces. Areas of congruent triangles are equal.
Area of one face = Area of the equilateral triangle with side = 5cm
= $\frac{\sqrt{3}a^{2}}{4}$
= $\frac{\sqrt{3}}{4}\times 5^{2}$
= 10.83 sq.cm   (rounded to the hundredth)
Hence the surface area of the tetrahedron = 4 x area of one face = 4 x 10.83 sq.cm = 43.3 sq.cm.