document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1192032>Question 1192032</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Consider a regular pentagon. Find the measures of the following in degrees.<BR>\n" );
document.write( "(a)<BR>\n" );
document.write( "each interior angle of a regular pentagon<BR>\n" );
document.write( " °<BR>\n" );
document.write( "(b)<BR>\n" );
document.write( "each exterior angle of a regular pentagon<BR>\n" );
document.write( " °<BR>\n" );
document.write( "Consider extending the sides of the regular pentagon to create a regular pentagram. Find the measures of the following in degrees.<BR>\n" );
document.write( "(a)<BR>\n" );
document.write( "each acute interior angle of a regular pentagram<BR>\n" );
document.write( " °<BR>\n" );
document.write( "(b)<BR>\n" );
document.write( "each reflex interior angle of a regular pentagram<BR>\n" );
document.write( " ° </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #823930 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=823930\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Problem 1, Part (a)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Draw a regular pentagon with all five sides the same length, and all interior angles the same.<BR>\n" );
document.write( "Plot a point at the center of the pentagon.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "From this center point, extend segments to the five vertices. This forms 5 isosceles triangles that are identical copies of each other.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The vertex angle is 360/n = 360/5 = 72 degrees.<BR>\n" );
document.write( "The base angle is (180-72)/2 = 108/2 = 54<BR>\n" );
document.write( "Two adjacent triangles have their adjacent base angles combine to form a single interior angle. So 2*54 = 108 is the interior angle measure.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "An alternative approach is to use this formula<BR>\n" );
document.write( "i = measure of interior angle<BR>\n" );
document.write( "i = 180(n-2)/n<BR>\n" );
document.write( "i = 180(5-2)/5<BR>\n" );
document.write( "i = 180(3)/5<BR>\n" );
document.write( "i = 540/5<BR>\n" );
document.write( "i = 108\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Answer: Each interior angle of a regular pentagon is 108 degrees.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "=============================================================\r<BR>\n" );
document.write( "\n" );
document.write( "Problem 1, Part (b)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "E = exterior angle of a regular polygon with n sides<BR>\n" );
document.write( "E = 360/n<BR>\n" );
document.write( "E = 360/5<BR>\n" );
document.write( "E = 72<BR>\n" );
document.write( "This is equal to the measure of the central angle found in the previous part.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Alternatively we can use the result of part (a)<BR>\n" );
document.write( "i = interior angle<BR>\n" );
document.write( "E = exterior angle<BR>\n" );
document.write( "i+E = 180<BR>\n" );
document.write( "E = 180 - i<BR>\n" );
document.write( "E = 180 - 108<BR>\n" );
document.write( "E = 72\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Answer: 72 degrees.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "=============================================================\r<BR>\n" );
document.write( "\n" );
document.write( "Problem 2, Part (a)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "This is what the diagram looks like when we extend the exterior sides of the pentagon to form a star shaped pentagram<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" width=\"50%\" src = \"https://i.imgur.com/ZuKuKwY.png\"><BR>\n" );
document.write( "Notice the angles 108 and 72 show up, which were calculated back in the previous problem.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Add up the angles of the triangle to solve for x.<BR>\n" );
document.write( "x+72+72 = 180<BR>\n" );
document.write( "x+144 = 180<BR>\n" );
document.write( "x = 180-144<BR>\n" );
document.write( "x = 36\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Answer: 36 degrees.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "=============================================================\r<BR>\n" );
document.write( "\n" );
document.write( "Problem 2, Part (b)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Erase the pentagon in the middle to get this<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" width=\"50%\" src = \"https://i.imgur.com/m9HTz9v.png\"><BR>\n" );
document.write( "The red angle y is the reflex interior angle we want.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "If we were to add the pentagon back in, and relevant angle measures<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" width=\"50%\" src = \"https://i.imgur.com/WV0XJ9q.png\"><BR>\n" );
document.write( "Then it's probably very clear that the red angle is:<BR>\n" );
document.write( "y = 72+108+72<BR>\n" );
document.write( "y = 252\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Answer: 252 degrees<BR>\n" );
document.write( "</font> </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );