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

In quadrilateral ABCD, AB = BC and AD = CD.

In ∆ABD and ∆CBD,

AB = CB (Given)

BD = BD (Common)

AD = CD (Given)

∴ ∆ABD ≅ ∆CBD (SSS congruence criterion)

So, ∠ABD = ∠CBD .....(1) (CPCT)

∠ADB = ∠CDB .....(2) (CPCT)

From (1) and (2), we conclude that

BD bisects both ∠ABC and ∠ADC.

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