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Let the diagonals intersect at O.

Rhombus =>  AB = BC = CD = DA and diagonals AC and BD are perpendicular.

compare the triangles ABO and  BOC.
         angle AOB = 90 = angle AOC.
          side  AB = side AC
         side  OB is common.
 hence they are congruent.      So   angle  ABO = angle CBO 
      hence diagonal BD bisects angle B.

You will find similarly,  AOD and DOC are congruent.   There is one common side, one right angle and AD = CD.  Hence OD (or BD) bisects angle D.

Similarly  triangles  BOC and COD are congruent. Hence OC (or AC) bisects angle C.

similarly, from triangles DOA and AOB , we find OA (or CA)  bisects angle A