SOLUTION: In Trapezium PQRS, the side PQ and SR are parallel. Angle RSP is 120 and RS=SP= 1/3 PQ. what is the size of angle PQR.

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Question 1069908: In Trapezium PQRS, the side PQ and SR are parallel. Angle RSP is 120 and RS=SP= 1/3 PQ. what is the size of angle PQR.
Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
Since PQ and RS are parallel,
angles RSP and SPQ are supplementary.
So, the measure of angle SPQ is
180%5Eo-120%5Eo=60%5Eo .
Below I drew PQRS .
Since RS=PS=%281%2F3%29PQ ,
I added two extra points on PQ
points T and U ,
so that RS=PS=PT=TU=UQ .
I also added some green lines
connecting the new points to R and S .

As you see in the drawing,
there are three equilateral triangles,
and all green sides are congruent with
RS , PS , PT , TU , and UQ .
That makes triangle QUR isosceles.
Since vertex angle QUR is supplementary
to 60%5Eo angle TUR ,
angle QUR measures 180%5Eo-60%5Eo=120%5Eo ,
and the base angles, PQR=UQR and QRU ,
measure %28180%5Eo-120%5Eo%29%2F2=highlight%2830%5Eo%29 .

MORE DETAILED EXPLANATION:
Triangle PTS is an equilateral triangle,
because it has congruent sides PT and PS
flanking a 60%5Eo angle.
That makes TS=PS=%22...%22 ,
and makes the measures of angles PST and TSR%7D%7D+%7B%7B%7B60%5Eo .
That,in turn, makes triangle RST equilateral.
You can figure out that triangle TUR
is equilateral too.