document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1189508>Question 1189508</A>: <I><SPAN STYLE=\"word-break: break-all;\"> In the diagram below, Angle DCE=Angle ACB, Angle BCF is a right angle, and CE ⊥ AB. Find the measure of Angle BAD\r<BR>\n" );
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document.write( "Diagram: https://imgur.com/a/enff97E </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #820960 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=820960\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Let x be the measure of angle DCE. \r<BR>\n" );
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document.write( "Because Angle DCE = Angle ACB, this means ACB is also x.\r<BR>\n" );
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document.write( "Extend out segment CE so it's a longer line. Make sure this longer line intersects AB. Label this intersection point G.\r<BR>\n" );
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document.write( "Note in the diagram that angles DCE and ACG are vertical angles. This means angle ACG is also x.\r<BR>\n" );
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document.write( "We're told that angle BCF is a right angle (aka 90 degrees). If we knew what angle FCG was, then we could solve for x.\r<BR>\n" );
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document.write( "Angle FBC is 23 degrees in the diagram.\r<BR>\n" );
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document.write( "Focus on right triangle FBC.<BR>\n" );
document.write( "Interior angle B is 23 degrees.<BR>\n" );
document.write( "The other acute angle of this right triangle must be 90-23 = 67 degrees. <BR>\n" );
document.write( "This is angle F of triangle FBC.<BR>\n" );
document.write( "More broadly, this is angle CFA.\r<BR>\n" );
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document.write( "Now focus your attention on triangle CFG. <BR>\n" );
document.write( "This is a right triangle due to CE ⊥ AB, ie the segments are perpendicular.\r<BR>\n" );
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document.write( "For triangle CFG, we found F = 67 earlier. This means C = 90-F = 90-67 = 23 degrees\r<BR>\n" );
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document.write( "Angle FCG = angle ABC = 23 degrees\r<BR>\n" );
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document.write( "We now have enough info to solve for x.<BR>\n" );
document.write( "Note that the angles FCG, GCA, and ACB are 23, x and x respectively. They combine to the angle BCF = 90.\r<BR>\n" );
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document.write( "So,<BR>\n" );
document.write( "(angleFCG) + (angleGCA) + (angleACB) = angle BCF<BR>\n" );
document.write( "(23) + (x) + (x) = 90<BR>\n" );
document.write( "2x+23 = 90<BR>\n" );
document.write( "2x = 90-23<BR>\n" );
document.write( "2x = 67<BR>\n" );
document.write( "x = 67/2<BR>\n" );
document.write( "x = 33.5\r<BR>\n" );
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document.write( "For the last part, we'll focus entirely on triangle ABC\r<BR>\n" );
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document.write( "We're given that B = 23 from the diagram.<BR>\n" );
document.write( "We just found that C = 33.5 which was the measure of x.\r<BR>\n" );
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document.write( "A+B+C = 180<BR>\n" );
document.write( "A+23+33.5 = 180<BR>\n" );
document.write( "A+56.5 = 180<BR>\n" );
document.write( "A = 180-56.5<BR>\n" );
document.write( "A = <font color=red>123.5</font> which is the measure of angle BAD.\r<BR>\n" );
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document.write( "Final Answer:<BR>\n" );
document.write( "<font color=red>123.5 degrees</font><BR>\n" );
document.write( "This value is exact.<BR>\n" );
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