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Diagonal AC of a parallelogram ABCD bisects ∠A. Show that(i) it bisects ∠C also,(ii) ABCD is a rhombus. |
Answer» Data: Diagonal AC of a parallelogram ABCD bisects ∠A. To Prove: (i) Diagonal AC also bisects ∠C. (ii) ABCD is a rhombus. Proof: i) Diagonal AC also bisects ∠A. ∴ ∠DAC = ∠BAC …………… (i) But, ∠DAC= ∠BCA (alternate angles)(ii) ∠BAC=∠DCA (alternate angles)(iii) From (i), (ii) and (iii), ∠BCA = ∠DCA ∴ AC bisects ∠C. (ii) ∠DAC = ∠DCA ∴ AD = DC But, AD = BC ∴ DC = AB ∴ AB = BC = CD = DA ∴ ABCD is a rhombus. |
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