Question

Diagonal AC of a parallelogram ABCD bisects ∠A.Show that

(i) It bisects ∠C also,

(ii) ABCD is a rhombus.


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Answer 1

Given:

• ABCD is a parallelogram.
• Diagonal :- AC

To Proof:

• It bisects ∠C also,
• ABCD is a rhombus.

Solution:

(i) ABCD is a parallelogram.

→∠DAC = ∠BCA (Alternate interior angles) … (1)

→ ∠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.

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(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.

Hence,

• AC bisects ∠C.
• ABCD is a rhombus.

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