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  • Diagonal AC of a parallelogram ABCD bisects ∠ A (see Fig. 8.19). Show that (ii) ABCD is a rhombus.

Q: 6        Diagonal AC of a parallelogram ABCD bisects \small \angle A (see Fig. \small 8.19). Show that

                (ii) ABCD is a rhombus.


 

Answers (1)

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Given: \angleDAC=\angleBAC ................1 

              \angleDAC=\angleBCA.................2  (Alternate angles)

             \angleBAC=\angleACD .................3  (Alternate angles)

From equation  1,2 and 3, we get

            \angleACD=\angleBCA...................4

From 2 and 4, we get

            \angleACD=\angleDAC

    In \triangle ADC,

                   \angleACD=\angleDAC       (proved above )

                        AD=DC         (In a triangle,sides opposite to equal angle are equal)

A parallelogram whose adjacent sides are equal , is a rhombus.

Thus, ABCD is a rhombus.

 

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