This probably seems like a super basic question, but I'm only on the level of an Honors Geometry course right now. Anyways, I don't understand why auxiliary lines are valid in proofs. Wouldn't they have to be included in the "given" when proving something geometric?
For example, for the "Triangle Sum Theorem" (The sum of the measures of the angles of a triangle is 180 degrees), step one in my textbook is:
Through B draw line BD paralel to line AC
This is an extra piece of information; without this line, we wouldn't be able to prove it.
Wouldn't it be like mentioning in proof X with a Triangle ABC, and casually adding in a step that Triangle ABC is isosceles, and as a reason listing "because I can draw it that way"?
Also, since this is my first question on the Mathematics Stackexchange forum, I'd appreciate any feedback on how to improve my next math quesiton. Thank you!