document.write( "Question 1102351: The perpendicular bisector of side AB of ∆ABC intersects the extension of side AC at D. Find the measure of ∠ABC if m∠CBD = 16° and m∠ACB = 118°. \n" ); document.write( "

Algebra.Com's Answer #717139 by greenestamps(13203) $\"\"$ $\"About$

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\n" ); document.write( "Let E be the midpoint of side AB, so that DE is the perpendicular bisector of AB.

\n" ); document.write( "Then triangle ABD is isosceles; and triangles AED and BED are congruent.

\n" ); document.write( "Let F be the point where BC and DE intersect.

\n" ); document.write( "Let x be the measure of angle ABC that we are looking for. Then

\n" ); document.write( "angle BFE is 90-x [complement of x]
\n" ); document.write( "angle BFD is 90+x [supplement of angle BFE]
\n" ); document.write( "angle BDF is 74-x [angle sum of triangle BDF, given that angle DBC is 16]

\n" ); document.write( "angle ACB is 118 [given]
\n" ); document.write( "angle BCD is 62 [supplement of ACB]
\n" ); document.write( "angle CFD is 90-x [vertical angle to BFD]
\n" ); document.write( "angle CDF is x+28 [angle sum of triangle CFD]

\n" ); document.write( "But angles BDF and CDF are corresponding angles in congruent triangles AED and BED, so
\n" ); document.write( " $\"74-x+=+x%2B28\"$
\n" ); document.write( " $\"46+=+2x\"$
\n" ); document.write( " $\"x+=+23\"$

\n" ); document.write( "The measure of angle ABC is 23 degrees.

\n" ); document.write( "There might well be easier ways to get to this result.... The above is what I came up with.
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