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 \(\angle RTU\) ?
+2863 
\(\text{Because angle TRS is 55, that means angle QRT is 125}\)
\(\text{We construct a line from Point R to our new point X. Let segment RX be parallel to QP and TU.}\)
\(\text{Angle PQR and angle QRX add up to 180. This means QRX must be 60}\)
\(\text{If angle QRT is 125 and angle QRX is 60, this means angle RTU is 65}\)
\(\text{Angle XRT and angle RTU sum up to 180. And since RTU is 65, this means angle RTU is }\boxed{115^{\circ}}\)
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+129899 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
