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in Quadrilaterals by (65.2k points)

ABCD is a rhombus. Show that diagonal AC bisects ∠A as well as ∠C and diagonal BD bisects ∠B as well as ∠D.

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ABCD is rhombus. Diagonal AC bisects ∠A and ∠C. 

To Prove: Diagonal BD bisects ∠B and ∠D. 

Proof: In a rhombus all sides are equal and opposite angles are equal to each other. 

In ∆ABC, AB = BC 

∴ ∠BAC = ∠BCA 

∠B AC = ∠DCA (alternate angles) 

∴ ∠BCA = ∠DCA …………… (i) 

∴ AC bisects ∠C. 

Now, ∠BCA = ∠DAC (alternate angles) 

∠BAC = ∠DAC 

∴ AC bisects ∠A . 

In ∆ABD, AD = DB 

∴ ∠ABD = ∠ADB 

∠ABD = ∠CDB 

∴ BD bisects ∠D. 

∠ADB = ∠CBD 

∠ABD = ∠CBD 

∴ BD bisects ∠B.

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