If Fig. PQRS is a quadrilateral and T and U are respectively points on PS and RS such that PQ = RQ.
Also, ∠ PQT = ∠ RQU and ∠ TQS = ∠ UQS. Then
QT = QU
∠QPT = ∠QRU
We have,
∠PQT = ∠RQU and, ∠TQS = ∠UQS
∴ ∠PQT + ∠TQS = ∠RQU + ∠UQS [Adding both angles]
⇒ ∠PQS = ∠RQS..........(i)
Thus, In Δ PQS and Δ RQS,
PQ = RQ [Given]
∠PQS = ∠RQS [From (i)]
and, QS = QS [Common side]
∴ Δ PQS ≅ Δ RQS [SAS congruence criterion]
⇒ ∠QPS = ∠QRS [C.P.C.T.C]
⇒ ∠QPT = ∠QRU............(ii)
Now, In Δ QPT and Δ QRU,
QP = QR [Given]
∠PQT = ∠RQU [Given]
∠QPT = ∠QRU [From (ii)]
∴ ΔQPT ≅ ΔQRU
⇒ QT = QU