Mathematics
Equilateral triangle ABD and ACE are drawn on the sides AB and AC of △ABC as shown in the figure. Prove that :
(i) ∠DAC = ∠EAB
(ii) DC = BE
Triangles
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Answer
(i) Given,
△ABD and △ACE are equilateral triangles.
⇒ ∠DAB = ∠ABD = ∠BDA = ∠EAC = ∠ACE = ∠CEA = 60°
From figure,
∠DAC = ∠DAB + ∠BAC
⇒ ∠DAC = 60° + ∠BAC …..(1)
∠EAB = ∠EAC + ∠BAC
⇒ ∠EAB = 60° + ∠BAC …..(2)
From eq.(1) and (2), we have :
⇒ ∠EAB = ∠DAC
Hence, proved that, ∠EAB = ∠DAC.
(ii) In △DAC and △BAE,
⇒ ∠EAB = ∠DAC (Proved above)
⇒ AD = AB (Sides of an equilateral triangle)
⇒ AC = AE (Sides of an equilateral triangle)
∴ △DAC ≅ △BAE (By S.A.S axiom)
∴ DC = BE (Corresponding parts of congruent triangles are equal)
Hence, proved that DC = BE.
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