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 \(\angle RTU\) ?

Guest Sep 14, 2019

#1**+2 **

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

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

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

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

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

.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

CPhill Sep 14, 2019