Need help with a proof

[OnyxEagle7]

So based on the attached image, I know that in order to prove this, I have to prove that line AD is larger than line AB. I see no concrete way to do this though, other than using the fact that since D is further from B than C, and dropping an altitude from A, you could use the Pythagorean theorem, but I can't assume that D is in that location right?
Thanks,
Onyx

 

[tkhunny]

Do you have the "Triangle Inequality"?

What if the length of CD were zero (0)?

 

[Dr.Peterson]

The givens should be explicit: Is it known that D is beyond C, or just that it is on line BC but not in segment BC, or merely that it is on line BC (in which case the claim is false)? We should not have to infer such facts from a picture.

 

[OnyxEagle7]

So my instructor confirmed that D is an extension of segment BC, but after thinking about it again, if BD>BC, then AD>AC because of the triangle inequality? does that make sense?

 

[pka]

So based on the attached image, I know that in order to prove this, I have to prove that line AD is larger than line AB. I see no concrete way to do this though, other than using the fact that since D is further from B than C, and dropping an altitude from A, you could use the Pythagorean theorem, but I can't assume that D is in that location right?
Thanks,
OnyxView attachment 13129

I have different take. Note that \(\displaystyle \Delta ABC\) is isosceles so \(\displaystyle m(\angle ACB)=m(\angle ABC)\).
A well known theorem: the measure of an exterior angle of a triangle is equal to the sum of the measures of the two opposite interior angles.
So we have \(\displaystyle m(\angle ACB)=m(\angle CDA)+m(\angle CAD)\).
What does that tell you.?

 

[OnyxEagle7]

thanks pka!
from that you can see that angle B is equal to angle D plus something, and so B must be larger
thanks again!