Question

Prove that if in any quadrilateral parts of diagonal are proportional then, it is a trapezium.

Answer 1

Heya. !!!!

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Given: ABCD is a quadrilateral and Diagonal AC and BD intersect at O such that AO/OC = DO/OB

 

In ∆AOD and ∆BOC

AO/OC = DO/OB. ( Given)

∠AOD =  ∠COB  (Vertically  opposite angles)

 

Thus ∆AOD ~ ∆BOC  (by SAS similarity creation)

⇒ ∠OAD = ∠OCB                ...  (1)

 

Now transversal AC intesect AD and BC such that ∠CAD = ∠ACB        (from (1))  (Alternate opposite angle)

 

So AD || BC

●●==>> Hence ABCD is a trapezium.●●


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Hope it helps u !!!!

# Nikky