QR is the common side for \$\triangle PQR\$ and \$\triangle SRQ\$, then prove \$\triangle PQR \cong \triangle SRQ\$.

MQ is the angle bisector of \$\angle NMP\$ and O is the mid-point of NP, then prove \$\triangle MON \cong \triangle MOP\$.

MNO is a equilateral triangle, P, Q and R are the mid-points of MN, NO and OM respectively and \$PQ ∥ NO \$. Prove \$\triangle MPQ \cong \triangle PNR\$.

\$\triangle ABC\$ and \$\triangle DEF\$ are isosceles triangles and have equal aeras. Prove \$\triangle ABC \cong \triangle DEF\$.

The two circles with centers A and B are orthogonal, CD is the common chord for the two circles, \$FE \bot CD\$ and I is the mid point of CD and FE. Prove \$\triangle CIE \cong \triangle DIF\$.