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प्रश्न
Diagonal AC of a parallelogram ABCD bisects ∠A (see the given figure). Show that
- It bisects ∠C also,
- ABCD is a rhombus
उत्तर
(i) ABCD is a parallelogram.
∴ ∠DAC = ∠BCA ...(Alternate interior angles) ...(1)
And, ∠BAC = ∠DCA ...(Alternate interior angles) ...(2)
However, it is given that AC bisects ∠A.
∴ ∠DAC = ∠BAC ...(3)
From equations (1), (2), and (3), we obtain
∠DAC = ∠BCA = ∠BAC = ∠DCA ...(4)
⇒ ∠DCA = ∠BCA
Hence, AC bisects ∠C.
(ii) From equation (4), we obtain
∠DAC = ∠DCA
∴ DA = DC ...(Side opposite to equal angles are equal)
However, DA = BC and AB = CD ...(Opposite sides of a parallelogram)
∴ AB = BC = CD = DA
Hence, ABCD is a rhombus.
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