Why are auxiliary lines valid in geometric proofs?

Question

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!

Answer 1

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"?

No. In your example, you add an extra hypothesis (the triangle was not supposed to be isoceles and then you say that it is isoceles, which isn't correct) whereas the textbook doesn't change the hypotheses of the theorem it proves by adding a new line. Moreover, as long as one could prove that this line exists under the hypotheses of the theorem we want to prove, then we are allowed to use it in the proof.

On the other hand, if proving the existence of the new line needed an extra hypothesis which was not included in the statement of the theorem, then in this case, you're not allowed to use this new line.

Answer 2

Through B draw line BD paralel to line AC

Such statements are really lemmas that haven't been proved. They basically assume that you can fill in the missing proof. In reality, it is really a way to hide the more obvious lemmas whos proof actually requires some subtle reasoning. A valid proof would, at least, require the following intermediate steps.

1 B is not on line $\overleftrightarrow{AC}$

$\qquad$ This is true since we are given $\triangle ABC$, which implies that points A, B, and C are not collinear.

2 There is a plane containing points A, B, and C

$\qquad$ There is a postulate for this.

3 There is a line in the plane containing points A, B, and C that passes through point B and is parallel to line $\overleftrightarrow{AC}$.

$\qquad$ This is a lemma whose proof depends on the Euclidean version of the parallel postulate and which book you are using.

4 There is a point, D, in the above line that is not point B.

$\qquad$ There is a postulate for this.

postscript

Generally the proofs of such constructions are assumed to be trivial enough that any reasonably educated mathematician can fill in the missing steps in his or her head. There is always the possibility that the statement made an unsupportable assumption. Euclid himself has made a few such mistakes. See for example the crossbar theorem.

Answer 3

In your particular case, drawing this auxiliary line is a valid move because of the Euclid's fifth postulate from his "Elements", which can also be restated as Playfair's axiom:

Through any point in the plane, there is at most one straight line parallel to a given straight line.

This postulate means that when you have a straight line (AC in your case, the base of your triangle), you can always draw another line parallel to it and passing through some point of choice (B in your case, the apex of your triangle), and there would be only one such line possible, so it will be unique (in simpler words: there's no way to screw it ;-) ).

This additional line doesn't change anything in your problem, but it can help you solve it by making some principles more apparent or making it possible to apply them (that's why it is called an auxiliary line). For example, in your case, it allows for finding a corresponding angle for each angle in your triangle and aligning them side by side so that it is more apparent that they add up to a straight angle (180°).

And that's the usual use of drawing auxiliary parallel lines: to transfer some angles to some other place in order to compare them or add them/subtract them. Sometimes it can also be used to produce similar triangles so that you could exploit the proportionality between their sides to find unknown lengths.

You can draw as many such lines as you please (of course less is more ;-) ), as long as they fit the conditions of the problem and doesn't change the problem into something else.

I encourage you to investigate other axioms and postulates of geometry too, because they can be useful for problem-solving as well.