PQR is a equilateral triangle and QRST is a square.prove that:-
- PT=PS
- angle SR=15 degree
Answer :
We have PQR is a equilateral triangle and QRST is a square ,
SO,
PQ = QR = PR ( sides of equilateral triangle )
And
QR = RS = ST = TQ ( Sides of square )
And
PRS = PRQ + QRS = 60 + 90 = 150 ---- ( 1 ) ( As we know all angle of equilateral triangle is 60 and all angles on a square is 90 )
And
PQT = PQR + RQT = 60 + 90 = 150 ---- ( 2 ) ( As we know all angle of equilateral triangle is 60 and all angles on a square is 90 )
From equation 1 and 2 , we get
PRS = PQT ---------------- ( 3 )
1 ) Now in PRS and PQT
PR = PQ ( Sides of equilateral triangle )
PRS = PQT ( From equation 3 )
And
RS = QT ( Sides of square )
Hence,
PRS PQT ( By SAS rule )
So,
PT = PS ( BY CPCT ) ( Hence proved )
2 ) Show PSR = 15
Now in PRS
PR = PS ,
So , By base angle theorem , we get
SPR = PSR
And we know angle sum properties in PRS
SPR + PSR + PRS = 180
PSR + PSR + 150 = 180 ( As we know PRS = 150 And SPR = PSR )
2 PSR = 180 - 150
2 PSR = 30
PSR = 15 ( Hence proved )
We have PQR is a equilateral triangle and QRST is a square ,
SO,
PQ = QR = PR ( sides of equilateral triangle )
And
QR = RS = ST = TQ ( Sides of square )
And
PRS = PRQ + QRS = 60 + 90 = 150 ---- ( 1 ) ( As we know all angle of equilateral triangle is 60 and all angles on a square is 90 )
And
PQT = PQR + RQT = 60 + 90 = 150 ---- ( 2 ) ( As we know all angle of equilateral triangle is 60 and all angles on a square is 90 )
From equation 1 and 2 , we get
PRS = PQT ---------------- ( 3 )
1 ) Now in PRS and PQT
PR = PQ ( Sides of equilateral triangle )
PRS = PQT ( From equation 3 )
And
RS = QT ( Sides of square )
Hence,
PRS PQT ( By SAS rule )
So,
PT = PS ( BY CPCT ) ( Hence proved )
2 ) Show PSR = 15
Now in PRS
PR = PS ,
So , By base angle theorem , we get
SPR = PSR
And we know angle sum properties in PRS
SPR + PSR + PRS = 180
PSR + PSR + 150 = 180 ( As we know PRS = 150 And SPR = PSR )
2 PSR = 180 - 150
2 PSR = 30
PSR = 15 ( Hence proved )