Given points $A, B, C$, and $D$ such that segments $A C$ and $B D$ intersect at point $E$. Segment $A E$ is 1 cm shorter than segment $A B$, $A E = D C$, $A D = B E$,

$\angle A D C = \angle D E C$. Find the length of $E C$.
$\angle B E A=\angle D E C=\angle A D C$, therefore triangles $A D C$ and $B E A$ are equal by two sides and the included angle. Thus, $A B=A C$. Therefore,

$E C=A C-A E=A B-A E=1$.

## Answer

1 cm.