# ABCD is a rhombus.show that diagonal AC bisects angle A as well as angle C and diagonal BD bisects angle B as well as angle D

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by vikramman

2014-11-16T02:58:53+05:30

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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