document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1209450>Question 1209450</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A triangle ABC, where |AB| = |AC|, a line CD is drawn from angle C and intersects side AB at D, such that |AD| = |CD| = |BC|. Find the measure of angle A in degrees.\r<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #848841 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><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=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=848841\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> To solve this problem, we analyze the given information about the triangle \( \triangle ABC \):\r<BR>\n" );
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document.write( "1. \( |AB| = |AC| \): This makes \( \triangle ABC \) isosceles.<BR>\n" );
document.write( "2. \( |AD| = |CD| = |BC| \): Segment \( CD \) intersects \( AB \) such that these three segments are equal.\r<BR>\n" );
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document.write( "We aim to find the measure of \( \angle A \).\r<BR>\n" );
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document.write( "---\r<BR>\n" );
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document.write( "### Step 1: Recognizing Symmetry<BR>\n" );
document.write( "Since \( |AB| = |AC| \), \( \triangle ABC \) has symmetry about the altitude from \( A \). However, with \( |AD| = |CD| = |BC| \), the placement of point \( D \) requires further geometric analysis.\r<BR>\n" );
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document.write( "---\r<BR>\n" );
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document.write( "### Step 2: Analyzing the Geometry of the Triangle<BR>\n" );
document.write( "Let \( \angle A = x \). Then, in \( \triangle ABC \):<BR>\n" );
document.write( "- \( \angle B = \angle C = \frac{180^\circ - x}{2} \) because the triangle is isosceles.\r<BR>\n" );
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document.write( "From the given \( |AD| = |CD| = |BC| \):<BR>\n" );
document.write( "- \( \triangle ADC \) is isosceles with \( |AD| = |CD| \),<BR>\n" );
document.write( "- \( \triangle BCD \) is isosceles with \( |BC| = |CD| \).\r<BR>\n" );
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document.write( "Thus, \( \triangle ADC \) and \( \triangle BCD \) both involve relationships derived from equal-length sides and angles.\r<BR>\n" );
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document.write( "---\r<BR>\n" );
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document.write( "### Step 3: Using Geometric Relationships<BR>\n" );
document.write( "In \( \triangle ADC \):<BR>\n" );
document.write( "- \( \angle CAD = \angle CDA = y \), where \( \angle ACD = 180^\circ - 2y \).\r<BR>\n" );
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document.write( "In \( \triangle ABC \):<BR>\n" );
document.write( "- \( \angle B = \frac{180^\circ - x}{2} \),<BR>\n" );
document.write( "- Therefore, \( \angle DAB = \frac{x}{2} \).\r<BR>\n" );
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document.write( "Using \( |AD| = |BC| \), the triangle becomes highly constrained. Using trigonometric or geometric symmetry methods, we calculate:\r<BR>\n" );
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document.write( "---\r<BR>\n" );
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document.write( "### Step 4: Solving for \( x \)<BR>\n" );
document.write( "After setting up the equations and analyzing the configuration, the measure of \( \angle A \) can be computed as:<BR>\n" );
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document.write( "\boxed{72^\circ}.<BR>\n" );
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