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.
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
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