Q. 64.4( 19 Votes )

# In the given figure, is a quadrilateral in which and Prove that

(i) bisects and

(ii)

(iii)

Answer :

Given: In ABCD, and

To prove: (i) bisects and

(ii)

(iii)

Proof:

(i) In ∆ABC and ∆ADC*,* we have,

AB = AD …given

BC = DC …given

AC = AC … common side

Hence, by SSS congruence rule,

∆ABC ≅ ∆ADC

∴ ∠BAC = ∠DAC and ∠BCA = ∠DCA …By cpct

Thus, AC bisects ∠A and ∠ C.

(ii) Now, in ∆ABE and ∆ADE*,* we have,

AB = AD …given

∠BAE = ∠DAE …from i

AE = AE …common side

Hence, by SAS congruence rule,

∆ABE ≅ ∆ADE

∴ BE = DE …by cpct

(iii) ∆ABC ≅ ∆ADC from ii

∴ ∠ABC = ∠ADC …by cpct

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