Question

If Fig. PQRS is a quadrilateral and T and U are respectively points on PS and RS such PQ = RQ.
$\angle$ PQT = $\angle$ RQU and $\angle$ TQS = $\angle$ UQS. Prove that QT = QU.
[4 MARKS]

Answer 1

Concept : 1 Mark
Process : 2 Marks
Proof : 1 Mark

We have,

$\angle$PQT = $\angle$RQU and, $\angle$TQS = $\angle$UQS

$\therefore$ $\angle$PQT + $\angle$TQS = $\angle$RQU + $\angle$UQS [Adding both angles]

$\Rightarrow$ $\angle$PQS = $\angle$RQS..........(i)

Thus, In $\Delta$ PQS and $\Delta$ RQS,
PQ = RQ [Given]
$\angle$PQS = $\angle$RQS [From (i)]
and, QS = QS [Common side]

$\therefore$ $\Delta$ PQS $\cong$ $\Delta$ RQS [SAS congruence criterion]

$\Rightarrow$ $\angle$QPS = $\angle$QRS [C.P.C.T.C]

$\Rightarrow$ $\angle$QPT = $\angle$QRU............(ii)

Now, In $\Delta$ QPT and $\Delta$ QRS,
QP = QR [Given]
$\angle$PQT = $\angle$RQU [Given]
$\angle$QPT = $\angle$QRU [From (ii)]

$\therefore$ $\Delta$QPT $\cong$ $\Delta$QRU
$\Rightarrow$ QT = QU