In the adjoining figure, AQRS is an equilateral triangle. Prove that,�

1.	In the adjoining figure, AQRS is an equilateral triangle. Prove that, i. arc RS ≅ arc QS ≅ arc QR ii. m(arc QRS) = 240°.
Answer» Proof: i. ∆QRS is an equilateral triangle, [Given] ∴ seg RS ≅ seg QS ≅ seg QR [Sides of an equilateral triangle] ∴ arc RS ≅ arc QS ≅ arc QR [Corresponding arcs of congruents chords of a circle are congruent] ii. Let m(arc RS) = m(arc QS)= m(arc QR) = x m(arc RS) + m(arc QS) + m(arc QR) = 360° [Measure of a circle is 360° , arc addition property] ∴ x + x + x = 360° ∴ 3x = 360° ∴ x = 360°/3 = 120° ∴ m(arc RS) = m(arc QS) = m(arc QR) = 120° (i) Now, m(arc QRS) = m(arc QR) + m(arc RS) [Arc addition property] = 120° + 120° [From (i)] ∴ m(arc QRS) = 240°

In the adjoining figure, AQRS is an equilateral triangle. Prove that, i. arc RS ≅ arc QS ≅ arc QR ii. m(arc QRS) = 240°.

Answer»

Proof:

i. ∆QRS is an equilateral triangle, [Given]

∴ seg RS ≅ seg QS ≅ seg QR [Sides of an equilateral triangle]

∴ arc RS ≅ arc QS ≅ arc QR [Corresponding arcs of congruents chords of a circle are congruent]

ii. Let m(arc RS) = m(arc QS)= m(arc QR) = x

m(arc RS) + m(arc QS) + m(arc QR) = 360° [Measure of a circle is 360° , arc addition property]

∴ x + x + x = 360°

∴ 3x = 360°

∴ x = 360°/3 = 120°

∴ m(arc RS) = m(arc QS) = m(arc QR) = 120° (i)

Now, m(arc QRS) = m(arc QR) + m(arc RS) [Arc addition property]

= 120° + 120° [From (i)]

∴ m(arc QRS) = 240°

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