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Prove that every diagonal of a rhombus bisects the angles at the vertices.

Rectilinear Figures

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From figure,

Prove that every diagonal of a rhombus bisects the angles at the vertices. Rectilinear Figures, ML Aggarwal Understanding Mathematics Solutions ICSE Class 9.

In △AOD and △COD,

AD = CD (All sides of rhombus are equal)

DO = OD (Common sides)

AO = OC (Diagonals of rhombus bisect each other)

∴ △AOD ≅ △COD (By SSS rule of congruency)

∴ ∠ADO = ∠CDO (By C.P.C.T.)

∴ BD bisects ∠D.

In △AOB and △COB,

AB = BC (All sides of rhombus are equal)

BO = OB (Common sides)

AO = OC (Diagonals of rhombus bisect each other)

∴ △AOB ≅ △COB (By SSS rule of congruency)

∴ ∠ABO = ∠CBO (By C.P.C.T.)

∴ BD bisects ∠B.

In △AOB and △AOD,

AB = AD (All sides of rhombus are equal)

AO = OA (Common sides)

OD = OB (Diagonals of rhombus bisect each other)

∴ △AOB ≅ △AOD (By SSS rule of congruency)

∴ ∠OAD = ∠OAB (By C.P.C.T.)

∴ AC bisects ∠A.

In △BOC and △DOC,

BC = CD (All sides of rhombus are equal)

OC = CO (Common sides)

OD = OB (Diagonals of rhombus bisect each other)

∴ △BOC ≅ △DOC (By SSS rule of congruency)

∴ ∠OCD = ∠OCB (By C.P.C.T.)

∴ AC bisects ∠C.

Hence, proved that every diagonal of a rhombus bisects the angles at the vertices.

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