Question

in fig,ABCD is a quadrilateral in which AB=AD and BC=CD.Prove that AC bisects angle A and angle C​

Answer 1

Step-by-step explanation:

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

(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