In triangle ABC points D and E are on bc such that BD=EC and AD=AE prove that AB=AC

In triangle ADE , 

AD = AE  [given]

⇒ ∠AED = ∠ADE  [angles opposite to equal sides are equal]

⇒ 180 - ∠AED = 180 - ∠ADE

⇒∠AEC = ∠ADB  ............(1)

in Δ ABD and Δ ACE

 AD = AE    [ given ]

∠ADB = ∠AEC  [ using (1) ]

BD = CE  [ GIVEN ]

Δ ABD is congruent to Δ ACE  [ SAS ]

⇒ AB = AC  [ CPCT ]

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In triangle ADE , angle ADE = angle AED (By isosceles triangle theprm)

angle ADB= 180-angle ADE

angle AEC=180- ANGLE AED

as, ade= aec

angle ADB= angle AEC

now, in triangle ABD triangle ACE

BD=CE(Given)

AD=AE(Given)

angle ADB = angle AEC

tharefore, triangle ABD is congruent to Triangle ACE

By cpct ab=ac

hence , prooved

  • 15

thank you so much

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