SOLUTION: Draw the diagram and set up the proof. "The bisector of the vertex angle of an isosceles triangle is parallel to the base."

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Question 23420: Draw the diagram and set up the proof.
"The bisector of the vertex angle of an isosceles triangle is parallel to the base."




Answer by venugopalramana(3286)   (Show Source): You can put this solution on YOUR website!
THE PROBLEM IS NOT WORDED PROPERLY.THE INTERNAL BISECTOR OF VERTICAL ANGLE IN AN ISOCELLES TRIANGLE WILL MEET THE BASE AT RIGHT ANGLES AND IS NOT PARALLEL TO THE BASE.HOWEVER THE EXTERNAL BISECTOR OF THE VERTICAL ANGLE BEING PERPENDICULAR TO THE INTERNAL BISECTOR WILL BE PARALLEL TO THE BASE.
YOU CAN SEE IT FROM THE FOLLOWING FIGURE.

LET THE TRIANGLE BE ABC WITH VERTEX AT A AND AB=AC.LET AD be THE INTERNAL BISECTOR...IN TRIANGLES ABD AND ACD ,WE HAVE
AB=AC....GIVEN
AD=AD
ANGLE BAD = ANGLE CAD ...(AD IS INTERNAL BISECTOR)
HENCE THE 2 TRIANGLES ARE IDENTICAL OR CONGRUENT.
HENCE ANGLE ADB = ANGLE ADC
SINCE BDC = 180 ,A STRAIGHT ANGLE ,WE HAVE ANGLE ADB = ANGLE ADC =90
HENCE BC IS PERPENDICULAR TO AD,THE INTERNAL BISECTOR OF VERTEX ANGLE.
BUT THE EXTERNAL BISECTOR OF SAME ANGLE IS PERPENDICULAR TO INTERNAL BISECTOR.HENCE THE EXTERNAL BISECTOR OF VERTEX ANGLE IS PARALLEL TO BC OR BASE SINCE LINES (EXTERNAL BISECTOR AND BC) PERPENDICULAR TO SAME LINE (AD) ARE PARALLEL
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