Geometry!? Indirect proofs?

Question

k for geometry we have to do indirect proofs... ughhh! So... other websites make it all confusing and say.. "assume the negation of the statement to be proved. Then, deductive reasoning will lead to a contradiction: two statements that cannot both be true". Sorry but that's confusing! Does anyone want to help me without being too confusing... !? thanks so much... k the question i have to answer is...Write an indirect proof: Given: segment FM is a median of Triangle DEF. measure of angle 1

Answer 1

Okay, so angle 1 refers to ∠FMD and angle 2 refers to ∠FME. Just so you know, FM is NOT an altitude. An altitude is a line segment from a vertex to the (possibly extended) opposite side that is _perpendicular_ to the opposite side. But if it were perpendicular, then ∠FMD ≅ ∠FME and we were explicitly told that is not the case. FM is a median, which is a line segment from a vertex to the midpoint of the opposite side. The median _may_ be equal to the altitude, but usually isn't (if it is, then the triangle is isosceles).

Anyway, onto the proof, which is really quite simple: suppose DF≅EF. Since FM is a median, DM≅EM. Finally, FM≅FM by reflexivity, so DFM≅EFM by SSS. This means in particular that ∠FMD ≅ ∠FME by CPCTC. However, m∠FMD < m∠FME, so ∠FMD ≇ ∠FME. Therefore, our initial assumption that DF≅EF was wrong, and DF≇EF. Q.E.D.

The basic idea of indirect proof is this: suppose you wish to prove a statement X. You can think "well, if X wasn't true, then Y would happen, and I already know Y doesn't happen, so X has to be true." Here's a rather straightforward (and humorous) example:

Statement: there are no elephants in my sock drawer

Proof: Suppose there was an elephant in my sock drawer. Then my sock drawer would be at least as large as the elephant it contains. However, my sock drawer is not the size of an elephant, neither is it larger. Therefore there are no elephants in my sock drawer. Q.E.D.

Okay, that sounds a little trivial, but it illustrates the point that proof by contradiction is an inherently natural form of proof. We use it all the time to eliminate impossibilities, because they would lead to things that we KNOW are not true. And then, when you have eliminated the impossible, whatever remains, however improbable, must be the truth. That's really the whole deal - show that it's impossible for X not to be true, then conclude that X is true.

Answer 2

An indirect proof is started by stating the opposite of what you are trying to prove. In this case you would state OM is congruent to NP then go about showing that this could not be true. I could do more with a picture. I hope this helps though.