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In the diagram below, \(\overline{PQ}\) and  \(\overline{TU}\)are parallel, and Q, R, and S are collinear. If \(\angle PQR = 120^\circ\) and \(\angle TRS = 55^\circ\) , then what is the degree measure of \(\angle RTU\) ?

 



 

 Sep 14, 2019
 #1
avatar+1533 
+2

\(\text{Because angle TRS is 55, that means angle QRT is 125}\)

 

\(\text{We construct a line from Point R to our new point X. Let segment RX be parallel to QP and TU.}\)

 

\(\text{Angle PQR and angle QRX add up to 180. This means QRX must be 60}\)

 

\(\text{If angle QRT is 125 and angle QRX is 60, this means angle RTU is 65}\)

 

\(\text{Angle XRT and angle RTU sum up to 180. And since RTU is 65, this means angle RTU is }\boxed{115^{\circ}}\)

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 Sep 14, 2019
edited by CalculatorUser  Sep 14, 2019
 #2
avatar+103858 
+2

Thanks CU......here's another way

 

Draw   PX  perpendicular to TU such that   PX intersects  TU  at W

 

Then  PQRTW forms an irregular pentagon

 

And the sum of the interior angles of any pentagon =  540

 

And note that the measures of angles  QRT, TWP and QPW  =  125, 90  and 90

 

So......the  sum of angles PQR , QRT , RTW, TWP  and QPW  = 540....so

 

120  + 125 + RTW + 90 +  90  = 540     simplify

 

425 + RTW =  540

 

RTW  = 540 - 425  =  115   =  RTU

 

 

 

cool cool cool

 Sep 14, 2019
 #3
avatar+1533 
+2

Yes, CPhill's solution is also a way to get the answer!

CalculatorUser  Sep 14, 2019

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