Question

In the given figure PQRS is a square and SRT is an equilateral triangle. prove that 1.angle PST = angle QRT, 2.PT=QT,3.anhleQTR =15degree

Answer 1

see figure in your question
A)
in triangle SRT
ANGLE TSR = 60° =STR=TRS. (Equilateral triangle)
PQRS is a square so,
ANGLE PQR=QRS=RSP=SPQ= 90°

angle TRS + angle QRS=60°+90°=150°
& angle TSR + angle SRP=60°+ 90°=150°
so, angle TSP = angle TRQ. (each 150°)
B)
in triangle PST and TQR
TS=TR. (sides of EQUILATERAL TRIANGLE)
PS=QR. (sides of square)
angle TSP= angle TRQ ( proved above)
so,
triangle PST is congruent to triangle QRT
by CPCT
so PT=QT
C)
in triangle QRT
ANGLE QTR+ ANGLE TRQ + ANGLE RQT=180
(ANGLE SUM PROPERTY OF TRIANGLE)
ANGLE QTR=ANGLE RQT.
(ANGLE OPPOSITE TO EQUAL SIDES)
2(ANGLE QTR)+ 150°=180
2(QTR)=180-150
ANGLE QTR = 30/2= 15°