x+∠QSR+∠UST=180x+∠QSR+∠UST=180 (straight line =180) and ∠R+∠T=90∠R+∠T=90 (as PRT is a right angle)
(1) The length of line segment QR is equal to the length of line segment RS --> triangle QRS is isosceles --> ∠RQS=∠QSR=180−∠R2∠RQS=∠QSR=180−∠R2 (as ∠RQS+∠QSR+∠R=180∠RQS+∠QSR+∠R=180 --> 2∗∠QSR+∠R=1802∗∠QSR+∠R=180 --> ∠QSR=180−∠R2∠QSR=180−∠R2). Not sufficient.
(2) The length of line segment ST is equal to the length of line segment TU --> triangle UST is isosceles --> ∠SUT=∠UST=180−∠T2∠SUT=∠UST=180−∠T2. Not sufficient.
(1)+(2) x+∠QSR+∠UST=180x+∠QSR+∠UST=180 --> x+180−∠R2+180−∠T2=180x+180−∠R2+180−∠T2=180 --> x+360−(∠R+∠T)2=180x+360−(∠R+∠T)2=180 --> since ∠R+∠T=90∠R+∠T=90 --> x+360−902=180x+360−902=180 --> x=45x=45. Sufficient.
