Definitions and Notations

Definitions:

• Congruent: The word ”Congruent” refers to objects that have the same shape and size, regardless of their position. In geometry, it means corresponding sides and angles are equal. In numbers, it means having the same remainder when divided by a certain value.

• Congruent triangles: Congruent triangles are triangles that have the same shape and size. When two triangles are congruent, their corresponding sides and angles are equal in measure. In other words, the corresponding sides of congruent triangles have the same length, and their corresponding angles have the same measure.

• SSS Postulate: The SSS (Side-Side-Side) Postulate is a geometric postulate used in triangle congruence. It states that if three sides of one triangle are congruent to three sides of another triangle, then the two triangles are congruent.

• SAS Postulate: The SAS (Side-Angle-Side) Postulate is another geometric postulate used in triangle congruence. It states that if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the two triangles are congruent.

• AAS Postulate: The AAS (Angle-Angle-Side) Postulate is also used in triangle congruence. It states that if two angles and a non-included side of one triangle are congruent to two angles and the corresponding non-included side of another triangle, then the two triangles are congruent.

• ASA Postulate: The ASA (Angle-Side-Angle) Postulate is another postulate used in triangle congruence. It states that if two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the two triangles are congruent.

• Identity Congruence: Identity congruence refers to the fundamental property in congruence relations where any mathematical object (often numbers or geometric figures) is congruent to itself. In other words, an object is always congruent to itself by the identity property. In mathematical notation, this can be expressed as $a\cong a$ (read as "a is congruent to a").

• Symmetric Property of Congruence: The symmetric property of congruence states that if two objects are congruent to each other, then the reverse is also true: if $a$ is congruent to $b$, then $b$ is congruent to $a$. This property mirrors the symmetry in the relation of congruence. Mathematically, if $a\cong b$, then it also implies $b\cong a$.

• Transitive Property of Congruence: The transitive property of congruence asserts that if two objects are congruent to each other and one of them is congruent to a third object, then the first object is also congruent to the third object. In other words, if $a$ is congruent to $b$ and $b$ is congruent to $c$, then it implies that $a$ is congruent to $c$.

  Mathematically, if $a\cong b$ and $b\cong c$, then it implies $a\cong c$.

Notations:

• Triangle $\Delta$

• Congruence $\cong$

• Angle $\angle$

• Line Segment AB $\overline{AB}$

• Correspondence $\leftrightarrow$

• Length of line segment AB $m\overline{AB}$