# 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

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by vikramman

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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

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