pls.solve QUESTION NO 22
Q. PQRST is a regulaer pentagon. Find the values of x, y and z.

Dear Student,

Please find below the solution to the asked query:

Given :  PQRST is a regular pentagon and we know in regular pentagon all five sides are equal to each other and all five interior angles is at 108$°$. So

$\angle$ PQR  =  $\angle$QRS  =  $\angle$RST =  $\angle$ STP = $\angle$ TPQ =  108$°$                             --- ( 1 )

In $∆$ PQR , PQ =  QR , SIdes of regular pentagon PQRST , SO from base angle theorem we get :
$\angle$ QPR  =  $\angle$ QRP                          --- ( 2 )

And from angle sum property of triangle we get in $∆$ PQR :

$\angle$ QPR  +  $\angle$QRP  +  $\angle$PQR  =  180$°$ , Substitute values from equation 1 and 2 and get :

$\angle$ QPR +  $\angle$QPR + 108$°$ = 180$°$ ,

2 $\angle$ QPR = 72$°$ ,

2 x = 72$°$                                   , From given diagram

x  =  36$°$

And in $∆$ RST , RS=  ST , Sides of regular pentagon PQRST , So from base angle theorem we get :
$\angle$ STR  =  $\angle$ SRT                          --- ( 3 )

And from angle sum property of triangle we get in $∆$ RST :

$\angle$ STR  +  $\angle$ STR  +  $\angle$ RST =  180$°$ , Substitute values from equation 1 and 3  and get :

$\angle$ STR +  $\angle$ STR + 108$°$ = 180$°$ ,

2 $\angle$ STR = 72$°$ ,

$\angle$ STR = 36$°$ , Then from equation 3 we get 

$\angle$ STR  =  $\angle$ SRT = 36$°$                              --- ( 4 )

And

$\angle$ RTP + $\angle$ STR =  $\angle$ RTP , Now we substitute values from equation 1 and 4 and get :

$\angle$ RTP + 36$°$ =  108$°$ ,

$\angle$ RTP  = 72$°$ ,

z  =  72$°$ , From given diagram

And

$\angle$ TRP + $\angle$ SRT + $\angle$ QRP =  $\angle$ QRS , Now we substitute values from equation 1 and 4 and get :

$\angle$ TRP + 36$°$ + $\angle$ QRP=  108$°$ ,

$\angle$ TRP  + $\angle$ QRP= 72$°$ ,

$\angle$ TRP +  36$°$ = 72$°$                   (  As in triangle PQR x  = $\angle$ QRP  = 36$°$ )

$\angle$ TRP = 36$°$

y  =  36$°$ , From given diagram


Therefore,

x  = 36$°$  , y  = 36$°$ and z  = 72$°$                                       ( Ans )

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