Two geometric figures are congruent if one can be picked up, rotated or flipped, and placed directly over the other so that it covers it exactly and completely. This test is called the principle of superposition.
- Line Segments: Two line segments are congruent if and only if they have the exact same length ($AB \cong CD \iff \text{length}(AB) = \text{length}(CD)$).
- Angles: Two angles are congruent if and only if they have the exact same degree measure ($\angle A \cong \angle B \iff m\angle A = m\angle B$). Arm length does not matter!
- Circles: Two circles are congruent if and only if they have the same radius ($r_1 = r_2$).
- Squares: Two squares are congruent if they have the same side length.
- Rectangles: Two rectangles are congruent if their corresponding length and breadth are equal.
Example: An angle $\angle PQR = 45^\circ$ has arms $PQ = 3\text{ cm}$ and $QR = 4\text{ cm}$. Another angle $\angle XYZ = 45^\circ$ has arms $XY = 8\text{ cm}$ and $YZ = 10\text{ cm}$. Are the two angles congruent?
Reasoning: The size of an angle depends strictly on the amount of rotation between its arms, NOT the length of its arms.
Answer: Yes, they are congruent ($\angle PQR \cong \angle XYZ$) because both measure exactly $45^\circ$.
Two shapes can look identical in shape (like a passport photo and a poster photo), but if their sizes differ, they are similar, NOT congruent! Congruence requires same shape AND same size.
In car manufacturing, spare engine parts must be strictly congruent to original parts so they fit into the engine cylinder block with microscopic precision.