SOLUTION: ABC isosceles and CD is angle bisector of the vertex angle prove that BD = AD Thanks for your help

Algebra ->  Geometry-proofs -> SOLUTION: ABC isosceles and CD is angle bisector of the vertex angle prove that BD = AD Thanks for your help      Log On


   

Question 173153: ABC isosceles and CD is angle bisector of the vertex angle
prove that BD = AD
Thanks for your help
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
If you draw the triangle, you might get this:


Photobucket - Video and Image Hosting



Since the triangle is isosceles, this means that AC and BC are congruent (since an isosceles triangle has equal sides). So this is given.

In order to prove that BD = AD, we need to show that the two triangles (that form when the segment CD is drawn) are congruent. Once we've proven that the two triangles are congruent, we can easily show that the corresponding parts are congruent. 






      Statement           |                         Reason
------------------------------------------------------------------------
  
1.   AC = BC                                         Given 


2.   CD = CD                                         Reflexive Property of Congruence


3.   angle ACD = angle BCD                           Definition of Angle Bisector
  

4.   triangle ACD = triangle BCD                     SAS Postulate                   



5.   BD = AD                                         CPCTC




Notes


#1) Remember an "angle bisector" cuts an angle in half. So this means that the two halves of the angle are equal (in this case angles ACD and BCD )


#2) statement 4 uses statements 1, 2, and 3.


#3) CPCTC stands for "Corresponding parts of congruent triangles are congruent".