Concept Notes 1: The Principle of Superposition & Geometric Congruence
Two plane geometric figures are called congruent if they have the exact same shape and the exact same size.
1. The Superposition Principle & The Congruence Symbol ($\cong$)
- Superposition: If figure $F_1$ can be moved, rotated, or flipped so that it superimposes (covers exactly) figure $F_2$, then $F_1$ is congruent to $F_2$.
- The Symbol $\cong$:
- The tilde $\sim$ represents same shape.
- The equality $=$ represents same size.
- Combining them yields $\cong$ ("is congruent to").
Congruence in Simple Plane Figures:
- Line Segments: Two line segments are congruent if and only if they have the same length ($AB \cong CD \iff AB = CD$).
- Angles: Two angles are congruent if and only if they have the same degree measure ($\angle A \cong \angle B \iff m\angle A = m\angle B$).
- Circles: Two circles are congruent if and only if they have the same radius.
- Squares: Two squares are congruent if and only if they have the same side length.
2. Congruence of Triangles & One-to-One Correspondence
Two triangles $\triangle ABC$ and $\triangle PQR$ are congruent if their vertices can be matched in a one-to-one correspondence such that all 3 pairs of corresponding sides and all 3 pairs of corresponding angles are equal:
$$\begin{array}{ccc}
\text{Vertices} & \text{Sides} & \text{Angles} \\
A \leftrightarrow P & AB = PQ & \angle A = \angle P \\
B \leftrightarrow Q & BC = QR & \angle B = \angle Q \\
C \leftrightarrow R & CA = RP & \angle C = \angle R
\end{array}$$
Critical Naming Rule: The statement $\triangle ABC \cong \triangle PQR$ implies that $A$ corresponds to $P$, $B$ to $Q$, and $C$ to $R$. Writing $\triangle ABC \cong \triangle QPR$ is geometrically false unless the triangles are equilateral!