If I have a diagram, like the following:
And I want to make make a proof for something like how segment AB is $\cong$ to segment AC if segment BD $\cong$ segment DC (using Perpendicular Bisector Theorem), well to do this I would need to show that segment AD $\bot$ segment DC (or segment BD).
Can I state that it is a given that the two are $\bot$ because a right angle is shown (so this would be given), or do I need to say, first, that m$\angle$ADC = right $\angle$ (given), and then say that segment AD $\bot$ segment DC (def of $\bot$ lines)
So, If the right angle (the rectangle in yellow) is given. I think is okay to say that $AD\perp BC$. Now, you say that then $AB=AC$ iff $BD=BC$ by using the perpendicular bisector theorem.
Even though this is fine, I think this is overkilling since it feels that the thing you want to prove follows trivially from this theorem.
Another way to prove this is by using Pythagorean theorem. $BD^2+AD^2=AB^2$ and $ DC^2+AD^2=AC^2$. If you substract these two equalities, you get $BD^2-DC^2=AB^2-AC^2$. So, $BD=DC$ iff $AB=BC$.