Congruence of Triangles - Congruence of Plane Figures and Line Segments

Congruence is a fundamental geometric relationship where two objects have exactly the same shape and size. If two figures are congruent, they are identical in every physical dimension, and one can be placed over the other to cover it perfectly without any gaps or overlaps.

The method of superposition is used to check for congruence. This involves taking a trace-copy of one figure and placing it over another. If the two figures fit exactly on top of each other, they are said to be congruent. Visually, this means every point on the boundary and interior of the first figure matches a corresponding point on the second figure.

The symbol for congruence is $\cong$. It is a combination of the 'similar' symbol $(\sim)$ and the 'equal' symbol $(=)$. For instance, if figure $F_1$ is congruent to figure $F_2$, we write it mathematically as $F_1 \cong F_2$.

Two line segments are congruent if and only if they have the same length. For example, if line segment $AB$ measures $5$ cm and line segment $CD$ also measures $5$ cm, then $AB \cong CD$. Visually, they represent the same distance between two points, regardless of their orientation or position on a plane.

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 implies that the 'opening' between the two arms of the angles is identical.

Two squares are congruent if they have the same side length. Since all angles in any square are $90^\circ$, matching the side lengths ensures that the entire perimeter and area will match exactly upon superposition.

Two circles are congruent if they have the same radius. Since all circles share the same 'round' shape, the radius $r$ is the only parameter that determines their size. If $r_1 = r_2$, one circle will perfectly cover the other when their centers are aligned.