Congruence is the most often used Geometric Property in reasoning and proofs. The word congruence is used to indicate the combined relationship of same shape and same size between geometrical figures. The congruence relationship is defined for basic ideas like line segments and angles and used in extending the concept to other geometric shapes.
Congruence is an idea equivalent to equality (=) in numerical expressions.

Congruence denotes the total agreement between geometric figures both in shape and size. Two or more shapes are said to be congruent if they are described by identical values in all the dimensions defining them.

This means their shapes agree in definition and in measurement as well.
Congruence is denoted by the symbol ≅.
Congruent segments are segments that have the same measure. If segments AB and CD are congruent then the lengths AB and CD are equal.

ABCD   ⇔   AB = CD.

Congruent Segments

The congruent line segments are indicated by the same mark across the lines. Congruent segments can also be identified by the measures labeled over them.

In the above diagram, ABCD.         Both the segments are marked with |.
                                   PQRS.         Both the segments are marked with ||.
                                   JK ≅ LM           Both the segments measure 5 cm each.

Examples of Congruent segments:
  1. In an isosceles triangle the legs are congruent.
  2. In an equilateral triangle all the three sides are congruent.
  3. In a parallelogram the opposite sides are congruent.
  4. In a square or rhombus all the four sides are congruent.
  5. In a rectangle or a square, the diagonals are congruent.
  6. All the diameters of a circle are congruent.
  7. Radii of congruent circles are congruent.
  8. The mid point of a line segment divides the segment into two congruent segments.
Congruent angles are angles that have the same measure. If ∠A and ∠B are congruent, then m ∠A = m ∠B.

∠A ≅ ∠B  ⇔ m ∠a = m ∠B.

Congruent Angles

In the above diagram angles ABC and DEF are marked with identical marks.
∠ABC ≅ ∠DEF                   Angles are congruent
m ∠ABC= m ∠DEF             Measures of angles are equal.

The angle bisector divides the angle into two congruent angles.
Examples of congruent angles:
  1. All right angles are congruent.
  2. All straight angles are congruent.
  3. Vertical angles are congruent.
  4. All the three angles of an equilateral angles are congruent.
  5. The Base angles of an isosceles triangle are congruent.
  6. In a parallelogram opposite angles are congruent.
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Congruent circles are circles with congruent radii.
If two circles have congruent radii or diameters then, the two circles are congruent. Circle O ≅ Circle O'     ⇔   OA ⇔ O'A' where OA and O'A' are radii of the circles O and O' respectively.

Circumcircles of congruent shapes are congruent.

For example in the diagram given below, Δ ABC ≅ Δ DEF.
Their circumcircles are also congruent.

Congruent Circles

But the converse is not true. If two triangles are inscribed in congruent circles, the triangles need not be themselves congruent.
Triangles of same shape and size are congruent.
Corresponding parts of congruent triangles are congruent.  (CPCTC)
Each triangle has six parts, six angles and six sides. If all the corresponding parts of two triangles are congruent, then the two triangles are congruent.

Congruent Triangles

If Δ ABC is congruent to Δ DEF, the vertices of the two triangles correspond in the same order as the letters used to denote them.

Corresponding Parts

Congruence of triangles are reflexive, symmetric and transitive.
Δ1 ≅ Δ1                                                                 Reflexive
If Δ1 ≅ Δ2, the Δ2 ≅ Δ1                                          Symmetric
If Δ1 ≅ Δ2 and Δ2 ≅ Δ3, then Δ1 ≅ Δ3                    Transitive

The congruence postulates or theorems are used to prove congruence are given below:

1.  Side - Side - Side Congruence         (SSS)
     Two triangles are congruent, if the three sides of a triangle are congruent respectively to the corresponding sides of the other triangle.

2. Side - Angle - Side Congruence         (SAS)
    Two triangles are congruent if two sides and included angle of one of the triangle are respectively congruent to the two sides and included angle of the other triangle.

3. Angle - Side - Angle congruence       (ASA)
    Two triangles are congruent if two angles and the included side of one triangle are respectively congruent to two angles and the included side of the other triangle.

4. Angle - Angle - Side congruent          (AAS)
    Two triangles are congruent if two angles and a non included side of a triangle are respectively congruent to two angles and a non included side of the other triangle.

Proving triangle congruence and using CPCTC consequently are widely used in solving many geometrical problems.
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