Prove that a quadrilateral is a parallelogram if opposite sides are equal.
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Prove that a quadrilateral is a parallelogram if opposite sides are equal.
Don't spamm!!!❌
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Answer:
Converse of Theorem 1: If the opposite sides in a quadrilateral are equal, then it is a parallelogram. If AB = CD and BC = AD in the given quadrilateral ABCD, then it is a parallelogram. Given: The opposite sides in a quadrilateral ABCD are equal, AB = CD, and BC = AD. To Prove: ABCD is a parallelogram.
[tex]\large\green{\sf{Given~:~ABCD~is~quadrilateral~.}}[/tex]
[tex]\large\orange{\sf{To~proove~-~Its~a~parallelogram.}}[/tex]
[tex]\large\pink{\sf{Proof~÷~}}[/tex]
[tex]\large{\sf{BD~acts~as~transversal}}[/tex]
[tex]\large{\sf{BAC~=~∠DCA(Alternate~angles~
}}[/tex]
[tex]\large{\sf{In~ΔADC~and~ΔCBA,~we~have}}[/tex]
[tex]\large{\sf{AB~=~CD~(Given)}}[/tex]
[tex]\large{\sf{∠BAC~=~∠DCA~(Alternate~ angles)}}[/tex]
[tex]\large{\sf{AC~=~CA~(Common)}}[/tex]
[tex]\large{\sf{Hence,~by~SAS~rule,~we ~get ΔADC~≅~ΔCBA}}[/tex]
[tex]\large{\sf{DA~=B~C~(By~C.PC.T)}}[/tex]
Thus, Both the pair of opposite sides are equal in the quadrilateral ABCD, therefore ABCD is a parallelogram.
Hence we proved that a quadrilateral is a parallelogram if a pair of opposite sides are equal and parallel.
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