congruence, Inequality On sides of Triangles

Congruence, Inequality On sides of Triangles

Angle-Side-Angle(ASA) Congruence Criterion
    The Angle-Side-Angle (ASA) congruence criterion is a geometric principle used to establish the congruence of two triangles. This criterion states that if two triangles have two angles and the included side of one triangle equal to the corresponding two angles and included side of another triangle, then the two triangles are congruent. Here are some important facts about the ASA congruence criterion:

Criterion Statement:
    If in two triangles, one pair of angles and the included side are equal, respectively, to the corresponding angles and side of the other triangle, then the triangles are congruent.

Formal Notation:
    The ASA congruence criterion is often expressed symbolically as follows:
             
    If \( \Delta ABC\,\,and\,\,\Delta DEF \) are such that    
    Here, \( \left| \!{\underline {\, B \,}} \right. = \left| \!{\underline {\, E \,}} \right. ,\left| \!{\underline {\, C \,}} \right. = \left| \!{\underline {\, F \,}} \right. \) and \( BC = EF \)
    \( \therefore \Delta ABC \cong \Delta DEF \)

Construction of Congruent Triangles:
    The ASA criterion allows for the construction of congruent triangles. Given the two angles and the included side in one triangle, a corresponding triangle can be constructed with the same angles and side lengths.

Unique Solution:
    Similar to SAS, the ASA criterion typically results in a unique solution. If two triangles satisfy the ASA condition, there is only one way to match the angles and side such that the triangles are congruent.

Order of Elements:
    The order of the angles and side in the criterion matters. If \( \left| \!{\underline {\, B \,}} \right. = \left| \!{\underline {\, E \,}} \right. ,\left| \!{\underline {\, C \,}} \right. = \left| \!{\underline {\, F \,}} \right. \) and \( \overline {BC} = \overline {EF} \) in one triangle, it does not necessarily mean that \( \overline {AB} = \overline {DE} \) unless the corresponding angles and side are in the same order.

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

Application in Geometry Proofs:
    The ASA 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.