Two Basic Geometry Questions: Remind Me!

a) Can you give me an example of a postulate? And, can a
postulate be proven? If not, then how is it different from a
definition? You’ll recall a defintion cannot be proven; it is just accepted.

b) Side Splitting Theorem: IIRC, this is supposed to work only for right triangles. However, I saw an SAT question using similar triangles (which were not two right triangles). If you can picture two similar triangles drawn such that they share one common leg, the problem was set up like a side-splitting problem. So, my question is: Does the side-splitting theorem apply to ALL similar triangles?

Thanks, it’s been awhile…

  • Jinx :confused:

If only I could refresh my memory like I can refresh the screen!

A postulate may be true or false. A definition is shorthand, and has no truth value.

An example of a postulate would be “any two distinct points determine a line”. This could be demonstrated to be false by producing two distinct points that do not determine a line.

A definition is something like “two lines are parallel iff they do not intersect”. There may be no parallel lines, or every pair of lines may be parallel, but no matter what, the definition stands.

The thing about definitions is that they’re not necessary–you can’t prove anything with them that you can’t without. Postulates do change what you can prove.

I don’t know what the side-splitting theorem is, but if it deals only with the ratios of lengths of line segments, then it holds for any triangle if it holds for a right triangle. Perhaps you can enlighten me?

Never mind, I just looked it up. It does hold for all triangles, for the reason I gave above.

A postulate is a starting point for a proof. For instance, in your second question, the SAT question would start by postulating two similar triangles. Sure, in another proof, you might have to prove the triangles similar, but in this one, their similarity is stated as a postulate.

First you have your undefined terms – like point and line. Postulates are axioms. They are assumed to be true without proof – like two points establish a line. If you discover that two things you are calling points do not establish what you are calling a line, this does not mean that the postulate is false; it means that the things you are calling points and lines are not a model of your postulates.