ABCD is a rhombus. M and N are points on BD such that DN = MB. Prove that Triangle DNC is congruent to triangle BMC.

Hello!

It is well-known that in any parallelogram the opposite angles are equal. A rhombus is a parallelogram, so the angles and are congruent.

Also, it is known that any diagonal of a rhombus bisects the corresponding angles. Therefore angles and are congruent, and and are congruent. From this and the first paragraph we infer that the angles and are congruent (both are halves of the congruent angles).

Now we can finish the proof. The triangles and have two pairs of congruent sides: by the conditions and by definition of rhombus. Also the angles between these sides are equal in both triangles, as proved above. Therefore these triangles are congruent by the side-angle-side rule (SAS).