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Question

Suppose in a quadrilateral $$ABCD, AD = BC$$ and $$AC = BD$$. Prove that $$ABCD$$ is a trapezium.


Solution

Given: $$ABCD$$ is a quadrilateral,
$$AC = BD$$ and $$AD = BC$$

To prove: $$ABCD$$ is trapezium

Proof: In $$Δ ADB$$ and $$Δ BCA$$

(1) $$AD = BC$$ (Given)

(2) $$AC = BD$$ (Given)

(3) $$AB = AB$$ (Common side)

$$∴ Δ ADB ≅ Δ BCA$$ (SSS postulate)

$$∴ ∠A = ∠B$$ (Congruency property)

$$AC = BD$$ (Given)$$AD = BC$$ (Given)

Hence, $$ABCD$$ is an isosceles trapezium.

Mathematics

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