Congruence, Inequality On Sides of Triangles

Congruence, Inequality On Sides of Triangles

Side-Side-Side (SSS) Congruence Criterion 
    The Side-Side-Side (SSS) congruence criterion is a fundamental principle used in geometry to establish the congruence of two triangles. This criterion states that if the three sides of one triangle are equal to the corresponding three sides of another triangle, then the two triangles are congruent. Here are some important facts about the SSS congruence criterion:

Criterion Statement:
    If in two triangles, all three sides of one triangle are equal, respectively, to the corresponding three sides of the other triangle, then the triangles are congruent.

Formal Notation:
    The SSS congruence criterion is often expressed symbolically as follows:
                 
    If \( \Delta ABC\,\,and\,\,\Delta DEF \) are such that
    here, \( AB = DE,BC = EF,AC = DF \)
    \( \therefore \Delta ABC \cong \Delta DEF \)

Construction of Congruent Triangles:
    The SSS criterion allows for the construction of congruent triangles. Given the three sides of one triangle, a corresponding triangle can be constructed with the same side lengths.

Unique Solution:
    The SSS criterion typically results in a unique solution. If two triangles satisfy the SSS condition, there is only one way to match the sides such that the triangles are congruent.

Comparison with Other Criteria:
    The SSS criterion is distinct from other triangle congruence criteria, such as Angle-Side-Angle (ASA), Side-Angle-Side (SAS), and Angle-Angle-Side (AAS). It is essential to choose the appropriate criterion based on the given information in a particular problem.

Application in Geometry Proofs:
    The SSS congruence criterion is commonly used in geometric proofs where it is necessary to show that two triangles are congruent. By establishing congruence, additional properties and relationships between the triangles can be deduced.