+0

# In the diagram below,

0
110
3

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
+2 $$\text{Because angle TRS is 55, that means angle QRT is 125}$$

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

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

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

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

.
Sep 14, 2019
edited by CalculatorUser  Sep 14, 2019
#2
+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   Sep 14, 2019
#3
+2

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

CalculatorUser  Sep 14, 2019