document.write( "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. \n" ); document.write( "

Algebra.Com's Answer #685103 by KMST(5328) $\"\"$ $\"About$

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Since $\"PQ\"$ and $\"RS\"$ are parallel,
\n" ); document.write( "angles $\"RSP\"$ and $\"SPQ\"$ are supplementary.
\n" ); document.write( "So, the measure of angle $\"SPQ\"$ is
\n" ); document.write( " $\"180%5Eo-120%5Eo=60%5Eo\"$ .
\n" ); document.write( "Below I drew $\"PQRS\"$ .
\n" ); document.write( "Since $\"RS=PS=%281%2F3%29PQ\"$ ,
\n" ); document.write( "I added two extra points on $\"PQ\"$
\n" ); document.write( "points $\"T\"$ and $\"U\"$ ,
\n" ); document.write( "so that $\"RS=PS=PT=TU=UQ\"$ .
\n" ); document.write( "I also added some green lines
\n" ); document.write( "connecting the new points to $\"R\"$ and $\"S\"$ .
\n" ); document.write( "

\n" ); document.write( "As you see in the drawing,
\n" ); document.write( "there are three equilateral triangles,
\n" ); document.write( "and all green sides are congruent with
\n" ); document.write( " $\"RS\"$ , $\"PS\"$ , $\"PT\"$ , $\"TU\"$ , and $\"UQ\"$ .
\n" ); document.write( "That makes triangle $\"QUR\"$ isosceles.
\n" ); document.write( "Since vertex angle QUR is supplementary
\n" ); document.write( "to $\"60%5Eo\"$ angle $\"TUR\"$ ,
\n" ); document.write( "angle $\"QUR\"$ measures $\"180%5Eo-60%5Eo=120%5Eo\"$ ,
\n" ); document.write( "and the base angles, $\"PQR=UQR\"$ and $\"QRU\"$ ,
\n" ); document.write( "measure $\"%28180%5Eo-120%5Eo%29%2F2=highlight%2830%5Eo%29\"$ .
\n" ); document.write( "
\n" ); document.write( "MORE DETAILED EXPLANATION:
\n" ); document.write( "Triangle $\"PTS\"$ is an equilateral triangle,
\n" ); document.write( "because it has congruent sides $\"PT\"$ and $\"PS\"$
\n" ); document.write( "flanking a $\"60%5Eo\"$ angle.
\n" ); document.write( "That makes $\"TS=PS=%22...%22\"$ ,
\n" ); document.write( "and makes the measures of angles $\"PST\"$ and $\"TSR%7D%7D+%7B%7B%7B60%5Eo\"$ .
\n" ); document.write( "That,in turn, makes triangle $\"RST\"$ equilateral.
\n" ); document.write( "You can figure out that triangle $\"TUR\"$
\n" ); document.write( "is equilateral too.
\n" ); document.write( " \n" ); document.write( "

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