Question

ABCDE is a pentagon in which AB AE,
BC = ED and ZABC = ZAED.
(a) Prove that :
(i) AC = AD (ii) ZBCD = ZEDC
(b) If BC and ED are produced to meet at
X, prove that BX = EX.​

Answer 1

Given : ABCDE is a pentagon in which AB = AE,

BC = ED and ∠ABC = ∠AED.

To Find :  Prove that :

(i) AC = AD (ii) ∠BCD = ∠EDC

BC and ED are produced to meet at

X, prove that BX = EX.​

Solution:

in ΔAED and ΔABC

AE = AB   given

∠AED   = ∠ABC  given

ED = BC    given

=> ΔAED ≅ ΔABC   (SAS)

=> AD = AC

ΔAED ≅ ΔABC  

=> ∠ADE = ∠ACB

AD = AC

=> ∠ACD = ∠ADC

∠ACD + ∠ACB =  ∠ADC + ∠ADE

=> ∠BCD = ∠EDC

QED

BC and ED are produced to meet at  X

=> ∠XDC =  180° - ∠EDC

   ∠XCD =  180° - ∠BCD

=> ∠XDC =  ∠XCD

=> XC = XD

   BC  = DE   given

=> XC + BC = XD + DE

=> BX = EX

QED

Hence proved

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