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.

Answer 1

∠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.