Congruence of Triangles - Concept of Congruence

The Concept of Congruence: Congruence refers to the relation of two objects having the same shape and size. If two figures can be superimposed (placed one on top of the other) so that they cover each other exactly, they are said to be congruent. Visually, imagine two identical 1-rupee coins; they are congruent because they have the exact same dimensions and shape.

The Congruence Symbol: To denote congruence between two figures, we use the symbol $\cong$. This symbol conveys that the figures are both equal in size (represented by the $=$) and similar in shape (represented by the $\sim$).

Congruence of Plane Figures: Two plane figures $F_1$ and $F_2$ are congruent if the trace-copy of $F_1$ fits exactly on $F_2$. For example, two identical square tiles are congruent because one can be laid perfectly over the other to hide it completely.

Congruence of Line Segments: Two line segments are congruent if and only if they have the same length. If two segments $\overline{AB}$ and $\overline{CD}$ are both $5\text{ cm}$ long, they are congruent, written as $\overline{AB} \cong \overline{CD}$. Visually, if you place one segment on top of the other, their endpoints will coincide.

Congruence of Angles: Two angles are congruent if they have the same measure. For instance, if $\angle ABC = 60^\circ$ and $\angle PQR = 60^\circ$, then $\angle ABC \cong \angle PQR$. This means the degree of 'opening' between the two rays of each angle is identical, regardless of the length of the rays shown.

Congruence of Triangles: Two triangles are congruent if they are copies of each other and when superposed, they cover each other exactly. This implies that the three sides of one triangle are equal to the corresponding three sides of the other, and the three angles of one are equal to the corresponding three angles of the other. In $\Delta ABC \cong \Delta PQR$, vertex $A$ falls on $P$, $B$ on $Q$, and $C$ on $R$.

Correspondence in Triangles: In congruent triangles, the matching of vertices is called correspondence. For the statement $\Delta ABC \cong \Delta PQR$, the correspondence is $A \leftrightarrow P$, $B \leftrightarrow Q$, and $C \leftrightarrow R$. The order of letters in the congruence statement is vital; for example, if $A$ matches $P$, then $A$ must be in the same position in the name as $P$.

CPCT (Corresponding Parts of Congruent Triangles): This principle states that if two triangles are proved congruent, then all their corresponding parts (sides and angles) are equal. It is a standard justification used in geometric proofs after establishing congruence.