Mathematics
In the given figure, AB = AC. Prove that DECB is an isosceles trapezium.
Circles
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Answer
Given, AB = AC
So, ∠B = ∠C ……..(1) [As Angles opposite to equal sides are equal]
From figure,
DECB is a cyclic quadrilateral.
∴ ∠B + ∠DEC = 180° [Sum of opposite angles in cyclic quadrilateral = 180°]
⇒ ∠C + ∠DEC = 180° (Using 1)
But this is the sum of interior angles on one side of a transversal.
∴ DE || BC.
∴ ∠ADE = ∠B and ∠AED = ∠C [Corresponding angles]
Thus, ∠ADE = ∠AED
∴ AD = AE
⇒ AB - AD = AC - AE [As AB = AC]
⇒ BD = CE
Hence, we have DE || BC and BD = CE.
Hence, proved that DECB is an isosceles trapezium.
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