In the figure, AD = CD and AB = CB.  


(i) Prove that line BD bisects angle ADC (i.e. that line BD cuts angle ADC into two equal angles).  


(ii) Prove that line AC and line DB are perpendicular.



AnonymousConfusedGuy  Jan 29, 2018


AB  = CB

And DB  = DB


So...by  SSS, triangle DAB  is congruent to triangle DCB


But  this implies that  angle BDA  =  angle BDC


So...   BD bisects angle ADC


To prove the second part.....


Since  AD  = CD, then the angles opposite those sides in triangle ADC are also equal

So  angle DAC =  angle DCA



angle BDA  = angle BDC

angle DAC  =  angle DCA 

And  DE  = DE


So....by AAS, trianlgle   ADE  is congruent to triangle CDE


But this implies that  angle AED  =  angle CED

But...  m  AED  +  m CED  =  180

So, by substitution   m AED + m AED  = 180

2m AED  =  180       divide by 2 on both sides

m AED  = 90   =  m CED

And a line standing upon a line forming two right angles at their intersection means that the lines are perpendicular....so, .AC is perpendicular to DB 



cool cool cool

CPhill  Jan 29, 2018

Thank you so much!

AnonymousConfusedGuy  Jan 29, 2018

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