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Question
PQRS is a quadrilateral and T and U are points on PS and RS respectively such that PQ = RQ, ∠PQT = ∠RQU and ∠TQS = ∠UQS. Prove that QT = QU.
Solution
∠PQT = ∠RQU .....(i)
∠TQS = ∠UQS .....(ii)
Adding (i) and (ii)
∠PQS = ∠RQS
In ΔPQS and ΔRQS
∠PQS = ∠RQS
PQ = RQ ...(given)
QS = QS ...(common)
Therefore, ΔPQS ≅ ΔRQS ...(SAS criteria)
Hence, ∠QPS = ∠QRS
Now in ΔPQT and ΔRQU
∠QPS = ∠QRS
PQ = RQ ...(given)
∠PQT = ∠RQU ...(given)
Therefore, ΔPQT ≅ ΔRQU ...ASA criteria)
Hence, QT =QU.
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