In fig $3.38$ $△ Q R S$ is an equilateral triangle. Prove that,
A. $arc R S ≅ arc Q S ≅ arc Q R$
B. $m (arc Q R S) = 240^{\circ}$ .

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Solution

A. Given, $△ Q R S$ is equilateral. So, the three arcs $R S, Q S$ and $Q R$ subtend equal angles at the centre.
Hence, $m (arc Q S) = m (arc R S) = m (arc Q R) \dots (1)$ .

B. From $(1)$ :
$∠ Q O S = ∠ S O R = ∠ R O Q$ .
$∠ Q O S + ∠ S O R + R O Q = 360^{\circ}$ (complete circle)
$∴ ∠ Q O S = ∠ S O R = ∠ R O Q = \frac{360^{\circ}}{3} = 120^{\circ}$

As $m (arc Q S) = m (arc R S) = m (arc Q R)$
(equal arcs subtend equal angles at centre of the circle)
$⟹ m (arc Q S) = m (arc R S) = m (arc Q R) = 120^{\circ}$
$m (arc Q R S) = m (arc Q R) + m (arc R S)$
$= 120^{\circ} + 120^{\circ}$
$m (arc ∠ Q R S) = 240^{\circ}$

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