document.write( "Question 1163536: Each side of Traingle ABC is produced. AB extends till D, AC extends till F and BC extends till E. If AB=AC, BD=BE and AF=DF, find angle ABC. \n" ); document.write( "
Algebra.Com's Answer #788031 by greenestamps(13200)\"\" \"About 
You can put this solution on YOUR website!


\n" ); document.write( "A sketch (NOT to scale)....

\n" ); document.write( "AB=AC, so triangle ABC is isosceles;
\n" ); document.write( "BD=BE, so triangle BDE is isosceles;
\n" ); document.write( "AF=DF, so triangle AFD is isosceles.

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\n" ); document.write( "Let x be the measure of angle ABC.

\n" ); document.write( "Then x is the measure of angle ACB (isosceles triangle ABC).

\n" ); document.write( "Then the measure of angle ECF is also x (vertical angle).

\n" ); document.write( "Angle BAC is 180-2x (angle sum of triangle ABC).

\n" ); document.write( "Angle ADF is 180-2x (isosceles triangle AFD).

\n" ); document.write( "Angle BED is 180-2x (isosceles triangle DBE).

\n" ); document.write( "Angle CFE is x (angle sum of triangle CEF).

\n" ); document.write( "Angle AFD is 180-x (supplement to angle CFE).

\n" ); document.write( "Angle DBC is 180-x (supplement to angle ABC).

\n" ); document.write( "Now here is the sketch (still not to scale):

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\n" ); document.write( "Now look at quadrilateral DBCF. The angle sum is

\n" ); document.write( "FDB + DBC + BCF + CFD = (180-2x)+(180-x)+(180-x)+(180-x) = 720-5x

\n" ); document.write( "But the angle sum of any quadrilateral is 360 degrees:

\n" ); document.write( "\"720-5x+=+360\"
\n" ); document.write( "\"5x+=+360\"
\n" ); document.write( "\"x+=+72\"

\n" ); document.write( "ANSWER: The measure of angle ABC is 72 degrees.

\n" ); document.write( "Note that, with a measure of 72 degrees for angle ABC, the problem description defines part of the figure of a regular 5-pointed star.

\n" ); document.write( "Viewing the given not-to-scale sketch as part of a picture of a regular 5-pointed star makes it easy to see that all of the conditions of the problem as stated are satisfied.

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\n" ); document.write( "Comment to tutor @MathTherapy....

\n" ); document.write( "Nice, elegant solution, achieved by defining x to be the measure of angle BDE....
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