Question

$△$STR is an isosceles triangle where ST = TR.
$\angle$PSR = $\angle$QRS = 90${}^{\circ }$.

Which of these is correct?

• A. $△$PTS $\cong$ $△$QTR
• B. $△$PTS $≆$ $△$QTR
• C. SP = QT
• D. RQ = ST

Answer 1

The correct option is A $△$PTS $\cong$ $△$QTR

Given, in $△$STR,
ST = TR
$\angle$TSR = $\angle$TRS( opposite angles of equal sides are equal)
Given $\angle PSR=\angle QRS=90°$
$\angle PSR=\angle PST+\angle TSR$
$\angle QRS=\angle QRT+\angle TRS$
We know $\angle TSR=\angle TRS$
Hence $\angle PST=\angle QRT$---------(1)

Consider $△$PTS and $△$QTR,
ST = TR (equal sides of the isosceles $△$STR)
$\angle$PST = $\angle$QRT -------(from 1)
$\angle$PTS = $\angle$QTR (vertically opposite angles)

$△$PTS $\cong$ $△$QTR by ASA postulate.

So, the corresponding parts are :

SP = RQ
PT = QT
TS = TR