PQRS is a rhombus. XPQY is a straight line such that XP = PQ = QY . IF XS and YR are produced to meet at point Z, the measure of angle XYZ is  

 

Dear Student,

Please find below the solution to the asked query:

We form our diagram from given information , As :

Here PQ =  XP  =  QY =  QR  =  RS = SP                --- ( A ) ( As PQRS is a rhombus ) and PQ | | RS and QR | | SP 

Here , 

SPX =  PQR                  --- (1 ) ( Corresponding angle as QR | | SP and XY is transversal line )

And

YQR =  QPS                  --- ( 2 ) ( Corresponding angle as QR | | SP and XY is transversal line )

In XPS , XP  =  PS  from equation " A"  , So from base angle theorem we get 

PXS =  PSX                  --- ( 3 ) 

And from angle sum property of triangle we get in   XPS 

PXS +  PSX  +  SPX =  180° , Now substitute values from equation 1 and 3 we get

PXS  +  PXS  + PQR =  180°

2 PXS +  PQR =  180°

2 PXS = 180° PQR  

2 PXS = QPS                                      --- ( 4 )      ( We know adjacent angles are supplementary in rhombus )

And

In YQR , QY  =  QR  from equation " A"  , So from base angle theorem we get 

QYR =  QRY                                   --- ( 5 ) 

And from angle sum property of triangle we get in   YQR

QYR +  QRY +  YQR =  180° , Now substitute values from equation 2 and 5 we get

QYR +  QYR + QPS =  180°

2 QYR  + 2 PXS =  180°  , From equation 4

2 ( QYR +  PXS ) = 180°

QYR +  PXS = 90°

XYZ +  YXZ = 90°                                      --- ( 6 )    (  We know QYR =  XYZ and PXS = YXZ same angles )

Now from angle sum property of triangle we get in   XYZ

XYZ +  YXZ +  XZY =  180° , Now substitute values from equation 6 we get

90° + XZY =  180°

XZY =  90°                                                                  (  Ans )
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