SOLUTION: I can't seem to figue out a 2 column proof for C.4.4.2: The diagonols of an isosceles trapezoid are congruent.

Question 62185This question is from textbook elementary geometry
: I can't seem to figue out a 2 column proof for C.4.4.2: The diagonols of an isosceles trapezoid are congruent. This question is from textbook elementary geometry

Answer by venugopalramana(3286) (Show Source): You can put this solution on YOUR website!
I can't seem to figue out a 2 column proof for C.4.4.2: The diagonols of an isosceles trapezoid are congruent.
LET ABCD BE THE ISOCELLES TRAPEZOID WITH AB AT THE BOTTOM PARALLEL TO CD AT THE TOP.GIVEN AD=BC
TPT....AC=BD
CONSTRUCTION..DRAW CE PERPENDICULAR TO AB MEETING AB AT E.
DRAW DF PERPENDICULAR TO AB MEETING AB AT F.
EF IS ALONG AB||CD
HENCE AB||CD
ANGLE DFE = 90 = ANGLE CEF......BY CONSTRUCTION
HENCE DF||CE
HENCE DFEC IS RECTANGLE.SO...DF=CE AND DC=FE
IN TRIANGLES ADF AND CEB,WE HAV
AD=BC .GIVEN
ANGLE AFD= 90 = ANGLE CEB....BY CONSTRUCTION
DF = CE ...PROVED ABOVE
HENCE THE 2 TRIANGLES ARE CONGRUENT.
HENCE AF = EB
NOW IN RIGHT ANLED TRIANGLE AEC
AC^2=AE^2+CE^2=(AF+FE)^2+DF^2....[DF=CE..PROVED ABOVE.]
AC^2 = (EB+FE)^2+DF^2.........[AF=EB.... PROVED ABOVE]
AC^2 = BF^2+DF^2 = BD^2..IN THE RIGHT ANGLED TRIANGLE BDF.
AC=BD ...PROVED

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