Prove that an equilateral triangle is equiangular.

1.	Prove that an equilateral triangle is equiangular.
Answer» Given: ∆ABC is an equilateral triangle. To prove: ∆ABC is equiangular i.e. ∠A ≅ ∠B ≅ ∠C …(i) [Sides of an equilateral triangle] In ∆ABC, seg AB ≅ seg BC [From (i)] ∴ ∠C = ∠A (ii) [Isosceles triangle theorem] In ∆ABC, seg BC ≅ seg AC [From (i)] ∴ ∠A ≅ ∠B (iii) [Isosceles triangle theorem] ∴ ∠A ≅ ∠B ≅ ∠C [From (ii) and (iii)] ∴ ∆ABC is equiangular.

Answer»

Given: ∆ABC is an equilateral triangle.

To prove: ∆ABC is equiangular

i.e. ∠A ≅ ∠B ≅ ∠C …(i)

[Sides of an equilateral triangle]

In ∆ABC,

seg AB ≅ seg BC [From (i)]

∴ ∠C = ∠A (ii)

[Isosceles triangle theorem]

In ∆ABC,

seg BC ≅ seg AC [From (i)]

∴ ∠A ≅ ∠B (iii) [Isosceles triangle theorem]

∴ ∠A ≅ ∠B ≅ ∠C [From (ii) and (iii)]

∴ ∆ABC is equiangular.

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