SOLUTION: Ok, this one is killing me, I guess I just can't wrap my head around it. Tired of trying and failing to get it. Given: Triangle ACB is Isosceles with a base of segment AC. There

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Question 872841: Ok, this one is killing me, I guess I just can't wrap my head around it. Tired of trying and failing to get it. Given: Triangle ACB is Isosceles with a base of segment AC. There is a segment running from B to the base ending at D(not stated as a given, but is drawn). Given: Angle CBD is congruent to Angle ABD. Prove: D is the midpoint of segment AC. I have tried to write the two column proof and have no confidence I am doing anything write. This is the last homework question I have to complete for this semester. Any help with this is greatly appreciated.
Found 2 solutions by jim_thompson5910, mananth:
Answer by jim_thompson5910(35256) (Show Source):
You can put this solution on YOUR website!

Here is one way to do it

Number Statement Reason

1. Triangle ACB is Isosceles with a base of segment AC Given

2. Angle CBD is congruent to Angle ABD Given

3. AB = BC Definition of Isosceles Triangle

4. BD = BD Reflexive Property

5. Triangle ABD is congruent to Triangle CBD SAS Property Of Congruence

6. AD = DC CPCTC

7. D is the midpoint of AC Definition of Midpoint

Answer by mananth(16946) (Show Source):
You can put this solution on YOUR website!
In triangles CBD & ABD
angle CBD is congruent to angle ABD
side BC is congruent to side AB ( isosceles triangle)
AD is the common side
so the triangles are congruent
there fore AD = DC
therefore D is the mid point of AC
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OR you can also prove by angular bisector theorem
BD is the angular bisector of angle C
AC=BC
D is any point on the angular bisector of angle C
Any point on the angular bisector of and angle is equidistant from the endpoints of the sides of the angle contained by them
therefore D is the mid point of AB