Question 33947
Prove: If an isosceles triangle has an altitude from the vertex to the base, then the altitude bisects the vertex angle.
Given: Triangle ABC is isoceles;
AC=CB
 Segment CD is the altitude to base Segment AB...
WHERE D IS ON AB.
HENCE ANGLE CDA=90=ANGLE CDB
To Prove: Segment CD bisects angle C
THAT IS ANGLE DCA=ANGLE DCB
Plan: ??.
WE SHALL TRY TO PROVE BOTH TRIANGLES CDA AND CDB TO BE CONGRUENT
Proof:?? 
IN TRIANGLES CDA AND CDB WE HAVE 
AC=CB........GIVEN
CD=CD......SAME COMMON SIDE
ANGLE CDA=90=ANGLE CDB........GIVEN
HENCE THE 2 TRIANGLES ARE CONGRUENT BY THEOREM ON CONGUENCE OF RIGHT TRIANGLES
HENCE ANGLE DCA=ANGLE DCB