\$AB \bot DC\$, AB = DC and E is the mid-point of AB and DC. Then prove \$\triangle AED \cong \triangle CEB\$.

ABCDEF is a regular hexagon, prove \$\triangle FGA \cong \triangle CGD\$.

l, m and n are three lines intersecting at O, QR = PR. Prove \$\triangle QOR \cong \POR\$.

BG is the common side for \$\triangle AGB and \triangle CGB\$. Prove \$\triangle AGB \cong \triangle CBG\$.

Two circles with centers M and N are orthogonal, CD is the common chord for the two triangles and E is the mid-point of BA. Prove \$\triangle DBC \cong \triangle DAC\$