SOLUTION: line segmant ab is congruent to bc and cd and ad (that is abcd is a rhombus)prove that line segmant ac is perpendicular to line segmant bd (hint: use detour)

Algebra ->  Triangles -> SOLUTION: line segmant ab is congruent to bc and cd and ad (that is abcd is a rhombus)prove that line segmant ac is perpendicular to line segmant bd (hint: use detour)      Log On


   

Question 22975: line segmant ab is congruent to bc and cd and ad (that is abcd is a rhombus)prove that line segmant ac is perpendicular to line segmant bd (hint: use detour)
Answer by venugopalramana(3286) About Me  (Show Source):
You can put this solution on YOUR website!
LET ABCD BE THE RHOMBUS.WITH AB=BC=CD=DA.JOIN DIAGONALS AC AND BD. LET THEM MEET AT O.
IN TRIANGLES ABC AND ADC , WE HAVE
AB=AD...GIVEN
BC=DC...GIVEN
AC=AC....GIVEN
SO THEY ARE CONGRUENT.SO ANGLE BAC = ANGLE DAC ..
NOW CONSIDER TRIANGLES AOB AND AOD
AO=AO...COMMON SIDE
AB=AD...GIVEN
ANGLE BAC= ANGLE BAO=ANGLE DAC = ANGLE DAO...FROM ABOVE
SO THEY ARE CONGRUENT.
SO ANGLE AOD =ANGLE AOB ....BUT THEIR SUM IS 180 ...A STRAIGHT ANGLE
HENCE ANGLE AOD=ANGLE AOB=90..SIMILARLY ANGLE DOC = ANGLE ANGLE BOC=90