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Question
ABCD is a quadrilateral in which AB = BC and AD = CD. Show that BD bisects both the angles ABC and ADC.
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Solution
Given: ABCD is a quadrilateral in which AB = BC and AD = CD.
To show: BD bisects both the angles ABC and ADC.
Proof: Since, AB = BC ...(Given)
∴ ∠2 = ∠1 ...(i) [Angles opposite to equal sides are equal]
And AD = CD ...[Given]
⇒ ∠4 = ∠3 ...(ii) [Angles opposite to equal sides are equal]
On adding equations (i) and (ii), we get
∠2 + ∠4 = ∠1 + ∠3
⇒ ∠BCD = ∠BAD ...(iii)
In ΔBAD and ΔBCD,
AB = BC ...[Given]
∠BAD = ∠BCD ...[From equation (iii)]
And AD = CD ...[Given]
∴ ΔBAD ≅ ΔBCD ...[By SAS congruence rule]
Hence, ∠ABD = ∠CBD and ∠ADB = ∠CDB i.e., BD bisects the angles ABC and ADC. ...[By CPCT]
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