1.

In the given figure, ABCD is a quadrilateral in which AB = AD and BC = DC. Prove that(i) AC bisects ∠ A and ∠ C,(ii) BE = DE,(iii) ∠ ABC = ∠ ADC

Answer»

(i) Consider △ ABC and △ ADC

It is given that AB = AD and BC = DC

AC is common i.e. AC = AC

By SSS congruence criterion

△ ABC ≅ △ ADC ……… (1)

∠ BAC = ∠ DAC (c. p. c. t)

So we get

∠ BAE = ∠ DAE

We know that AC bisects the ∠ BAD i.e. ∠ A

So we get

∠ BCA = ∠ DCA (c. p. c. t)

It can be written as

∠ BCE = ∠ DCE

So we know that AC bisects ∠ BCD i.e. ∠ C

(ii) Consider △ ABE and △ ADE

It is given that AB = AD

AE is common i.e. AE = AE

By SAS congruence criterion

△ ABE ≅ ∠ ADE

BE = DE (c. p. c. t)

(iii) We know that △ ABC ≅ △ ADC

Therefore, by c. p. c. t ∠ ABC = ∠ ADC


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