Congruence, Inequatlity on Sides Of Triangles

Congruence, Inequatlity on Sides Of Triangles

Congruence in Circles:
    In the context of circles, congruence refers to the equality of two circles in terms of their radii. Two circles are congruent if they have the same radius.
    Congruence is a fundamental concept in geometry, providing a way to describe and compare geometric shapes in a precise manner. It is widely used in geometric proofs and constructions, helping mathematicians establish relationships between different figures and solve problems related to shape and size.

Congruence ( Definition ): Two geometrical figures are said to be congruent, if the have exactly the same shape.
Congruence of line segments  Two line segments are congrueant if and only if, their lengths are equal.
 
Here, AB = CD
Congruence of Angles: Two angles are congruent, if their measures are equal.
   
Here, \( m\left| \!{\underline {\, {BAC} \,}} \right. = m\left| \!{\underline {\, {EDF} \,}} \right. \)

Congruence of triangles: Two triangles are congruent if and only if one of them can be made to superpose on the other, so as to cover it exactly.
    NOTE (i) Two triangles are congruent if and only if there exists a correspondance between their vertices such that the corresponding sides and the corresponding angles of the two triangles are      equal or congruent.
    NOTE(ii) If \( \Delta ABC \) is congruent to \( \Delta DEF \) we write it as
                          \( \Delta ABC \cong \Delta DEF \)
    NOTE(iii)\( \left. \begin{gathered} \Delta ABC \cong \Delta DEF \Leftrightarrow \left| \!{\underline {\, A \,}} \right. = \left| \!{\underline {\, D \,}} \right. ,\,\,\,\,\,\,\,\,\, \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left| \!{\underline {\, B \,}} \right. = \left| \!{\underline {\, E \,}} \right. \,\, \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left| \!{\underline {\, C \,}} \right. = \left| \!{\underline {\, F \,}} \right. \, \hfill \\ \end{gathered} \right|\,\,\,\,\,\, \)