document.write( "Question 1209487: (31) If AB = BC = CD and ∠BCA = 54°, then find the measure of angle ABD.\r
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Algebra.Com's Answer #848916 by ikleyn(52781) $\"\"$ $\"About$

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\n" ); document.write( "(31) If AB = BC = CD and ∠BCA = 54°, then find the measure of angle ABD.\r
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document.write( "In triangle ABC, AB = BC (given). \r\n" );
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document.write( "It means that triangle ABC is isisceles, with base angles BAC and ACB of 54 degrees.\r\n" );
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document.write( "Hence, the angle ABC at the vertex is  180 - 54 - 54 = 180 - 108 = 72 degrees.\r\n" );
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document.write( "In triangle BCD, BC = CD (given).\r\n" );
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document.write( "It means that triangle BCD is isosceles.\r\n" );
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document.write( "Hence, its angles CBD and CDB are congruent, as the base angles.\r\n" );
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document.write( "The sum of these angles, CBD and CDB, is equal to the exterior angle ACD, which is 54 degrees.\r\n" );
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document.write( "Hence, angle CBD is half of 54 degrees, i.e. 27 degrees.\r\n" );
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document.write( "Now, angle ABD is the sum of angles ABC (72 degrees) and CBD (27 degrees)\r\n" );
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document.write( "So,  angle ABD = 72 + 27 = 99 degrees.\r\n" );
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document.write( "At this point, the problem is solved completely.\r\n" );
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document.write( "ANSWER.  Angle ABD is 99 degrees.\r\n" );
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