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

(i) bisects and


(ii)


(iii)


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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