document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=536514>Question 536514</A>: <I><SPAN STYLE=\"word-break: break-all;\"> how to i find the value of the angles PQR and QRS  with the only coordinates given for the parallelogram  P=(-2.1) , Q=(-6,4) and R = (4,3)  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #352486 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5328)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=352486\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> MY WAY<BR>\n" );
document.write( "If PQRS is a parallelogram the measures of angles PQR and QRS add up to 180 degrees.<BR>\n" );
document.write( "The sides QR and PS are congruent and parallel (slope of the lines are the same.<BR>\n" );
document.write( "Going from Q to R the x-coordinate increases by 10, while the y-coordinate decreases by 1. (slope =-1/10). Doing the same from P, we can locate S at (8,0).<BR>\n" );
document.write( "Consider the point N=(-2,0) and right triangle PSN.<BR>\n" );
document.write( "Tangent of angle PSN=1/10, so PSN measures 5.71 degrees<BR>\n" );
document.write( "Now consider point M=(4,0) and right triangle RSM.<BR>\n" );
document.write( "Tangent of angle RSM=3/4, so angle RSM (or angle RSN) measures 36.87 degrees.<BR>\n" );
document.write( "The measure of angle RSP can be calculated by difference as 31.16 degrees, and it's the same as the measure of angle PQR.<BR>\n" );
document.write( "The measure of supplementary angles QRS and SPQ, in degrees can be calculated as<BR>\n" );
document.write( "180-31.16=148.84<BR>\n" );
document.write( "WHAT MIGHT HAVE BEEN EXPECTED<BR>\n" );
document.write( "Where you studying law of cosines?<BR>\n" );
document.write( "If so, you were probably expected to calculate the lengths of the sides of triangle PQR and use those lengths and law of cosines to find the measure of PQR.<BR>\n" );
document.write( "So you would use the coordinates of the points and either plug into a formula for distance between points, or imagine a right triangle and apply Pythagoras to<BR>\n" );
document.write( "get<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=PQ=sqrt%28%28-2-%28-6%29%29%5E2%2B%281-4%29%5E2%29=sqrt%2825%29=5 ALIGN=MIDDLE  ALT=\"PQ=sqrt%28%28-2-%28-6%29%29%5E2%2B%281-4%29%5E2%29=sqrt%2825%29=5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=QR=sqrt%28%28-6-4%29%5E2%2B%284-3%29%5E2%29=sqrt%28101%29 ALIGN=MIDDLE  ALT=\"QR=sqrt%28%28-6-4%29%5E2%2B%284-3%29%5E2%29=sqrt%28101%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=RP=sqrt%28%284-%28-2%29%29%5E2%2B%283-1%29%5E2%29=sqrt%2840%29 ALIGN=MIDDLE  ALT=\"RP=sqrt%28%284-%28-2%29%29%5E2%2B%283-1%29%5E2%29=sqrt%2840%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Then, law of cosine says<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28sqrt%2840%29%29%5E2=5%5E2%2B%28sqrt%28101%29%29%5E2-2%2A5%2Asqrt%28101%29%2Acos%28PQR%29 ALIGN=MIDDLE  ALT=\"%28sqrt%2840%29%29%5E2=5%5E2%2B%28sqrt%28101%29%29%5E2-2%2A5%2Asqrt%28101%29%2Acos%28PQR%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=40=25%2B101-10%2Asqrt%28101%29%2Acos%28PQR%29 ALIGN=MIDDLE  ALT=\"40=25%2B101-10%2Asqrt%28101%29%2Acos%28PQR%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10%2Asqrt%28101%29%2Acos%28PQR%29=86 ALIGN=MIDDLE  ALT=\"10%2Asqrt%28101%29%2Acos%28PQR%29=86\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28PQR%29=86%2F10sqrt%28101%29=0.8573 ALIGN=MIDDLE  ALT=\"cos%28PQR%29=86%2F10sqrt%28101%29=0.8573\"   DISABLED_style= \"max-width:90%; height:auto;\" > ---> PQR=31.16 degrees<BR>\n" );
document.write( "NOTE: It is also possible that the answer was expected in radians, rather than degrees. My calculator says PQR=0.5408 when I switch the mode to radians.<BR>\n" );
document.write( "That would make QRS=3.1416-0.5408=2.6008 in radians.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28+400%2C+150%2C+-7%2C+9%2C+-1%2C+5%2C%0D%0A++grid%28+1+%29%2C%0D%0A++locate%28+-6%2C+4.8%2C+Q+%29%2C%0D%0Alocate%28+-2%2C+1.8%2C+P+%29%2C%0D%0Alocate%28+4%2C+3.8%2C+R+%29%2C%0D%0Alocate%28+8%2C+0.8%2C+S+%29%2C%0D%0Alocate%28+-2.3%2C+0.5%2C+N+%29%2C%0D%0Alocate%28+3.7%2C+0.5%2C+M+%29%2C%0D%0A++line%28+-2%2C+1%2C+-6%2C+4+%29%2C%0D%0A++line%28+-6%2C+4%2C+4%2C+3+%29%2C%0D%0A++line%28+4%2C+3%2C+8%2C+0+%29%2C%0D%0A++line%28+8%2C+0%2C+-2%2C+1+%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );