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If one angle of a triangle is equal to half the sum of the other two equal angles, then which type of triangle is this?

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Isosceles

Scalene

Equilateral

None

answer is D.

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Detailed Solution

Given one angle of a triangle is equal to half the sum of the other two equal angles.
Question Image

Let,

∠ A = \frac{1}{2} (∠ B + ∠ C)

By angle sum property of a triangle,
⇒

∠ A + ∠ B + ∠ C = 180^{o}

Substitute the value of

∠ A

\Rightarrow \frac{1}{2} (∠ B + ∠ C) + ∠ B + ∠ C = 180^{o}

\Rightarrow \frac{∠ B + ∠ C + 2 ∠ B + 2 ∠ C}{2} = 180^{o}

\Rightarrow \frac{3 ∠ B + 3 ∠ C}{2} = 180^{o}

\Rightarrow \frac{∠ B + ∠ C}{2} = \frac{180^{o}}{3}

\Rightarrow \frac{∠ B + ∠ C}{2} = 60^{o}

⇒

∠ A = 60^{o}

.
Thus, the given triangle is an acute angle triangle, which may or may not be an equilateral triangle as one angle is

60^{o}

.
Hence, the correct option is 4.

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