1. 6-3: Theorems on Lines and Planes -Indirect proof -Existence Proof -Uniqueness Proof Proof Geometry

2. What you need to be able to do: Understand and write a proof by contradiction Understand what uniqueness and existence mean (you do not need to be able to write these proofs). Understand math symbols used in logical reasoning. Understand and write a proof using an auxiliary set.

3. Looking Back… Indirect Proof of earlier Theorem If a line intersects a plane not containing it, then the intersection contains only one point. Given: Line L, plane E that does not contain L L intersects plane E at P Prove(by contradiction): The intersection contains only one point. Suppose: L intersects E in another point, Q (supposition) Then: If both P and Q lie in E then according to the Flat Plane Postulate, line L lies in E. (conclusion resulting from supposition) But: It is given that L does not lie in E (the CONTRADICTION) So: L intersect E only in point P

4. Given a line L and a point P not on the line, there is exactly one plane containing both of them. To prove this, we must prove two things: 1.There exists a plane E that contains P and L. i.Let Q and R be two points on line L. ii.The Plane Postulate states that any three points lie in at least one plane. So we know that there is a plane that contains points P,Q, and R, call this plane E. iii.The Flat Plane Postulate enables us to say that because Q and R are in plane E, that L is also in plane E. iv.So we now know that plane E contain P and L. This part of the proof is called an existence proof – we have to show that there is at least one plane R Q

5. R Q Given a line L and a point P not on the line, there is exactly one plane containing both of them. To prove this, we must prove two things: 2.This plane is the only one that contains P and L. (indirect proof) i.Suppose: there is another plane that contains P and L, call it plane E’. ii.Then: E’ contains P, Q, and R. iii.But: P, Q, and R are non-collinear and the Plane Postulate states that any three non- collinear points lie in exactly one plane iv.So: There can only be one plane that contains P and L

6. Notice how the proof splits into two parts: Existence: We had to show that there exists at least one plane through the point. It could be more than one. All we know for sure at this point is that one exists. Uniqueness: We then had to show that there is at most one plane that exists. Uniqueness proofs are usually by contradiction. If we prove both existence and uniqueness we have proven that there is exactly one. (AKA one and only one)

7. Existence Given two intersecting lines, there is exactly one plane containing them. Prove existence. Let Q be another point on L 1. Q cannot be on L 2 because two lines intersect in only one point. Using the previous theorem there is a plane E that contains L 2 and Q E contains P and Q so the Flat Plane Postulate helps us to say that it contains L 1

8. Uniqueness Given two intersecting lines, there is exactly one plane containing them. Prove uniqueness (by contradiction). Suppose: that another plane E’ contains L 1 and L 2 Then: E’ contains Q because Q is on L 1 But then both E and E’ contain Q and L 2 which contradicts the previous Theorem So: E is the only plane containing L 1 and L 2

9. Homework Pg. 185-186: # 2, 9-11.