1
Question
In the given figure, the points P, Q, R, S, and T lie on the circle. Prove that △OST is an isosceles triangle. [2 Marks]
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Solution
In the given figure. ∠POQ = ∠TOS (vertically opposite angle) ∠TOS = 70°
In the cyclic quadrilateral ORST, the sum of the opposite angles is 180°.
∠QRS - ∠OTS = 180°
110° -∠OTS = 180°
∠QTS = 180° - 110° = 70°
∠QTS = ∠OTS = 70°(same angle) [1 Mark]
In triangle OST,
∠TOS = 70°
∠OTS = 70°
∠TOS = ∠OTS
ST = OS ( sides opposite to equal angles are equal)
So the triangle is an isosceles triangle]
[1 Mark]
In the cyclic quadrilateral ORST, the sum of the opposite angles is 180°.
∠QRS - ∠OTS = 180°
110° -∠OTS = 180°
∠QTS = 180° - 110° = 70°
∠QTS = ∠OTS = 70°(same angle) [1 Mark]
In triangle OST,
∠TOS = 70°
∠OTS = 70°
∠TOS = ∠OTS
ST = OS ( sides opposite to equal angles are equal)
So the triangle is an isosceles triangle]
[1 Mark]
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