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.

Thanks!

AnonymousConfusedGuy
Jan 29, 2018

#1**+2 **

AD = CD

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

So

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

CPhill
Jan 29, 2018