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Unformatted text preview: 6-3 Theorems on Lines and Planes
A) Postulate 4-7 review
a. Postulate 4 (The Line Postulate)
i. For every 2 points there is exactly one line that contains both points b. Postulate 5
i. ii. Every plane contains at least three noncollinear points Space contains at least four noncoplanar points c. Postulate 6
i. If two points of a line lie in a plane, then the line lies in the same plane d. Postulate 7
i. Any three points lie in at least one plane, and any three noncollinear points lie in exactly one plane B) Prove theorem 3-2 with an Indirect Proof a. Theorem 3-2 i. If a line intersects a plane not containing it, then the intersection contains only one point
1. ii. Hypothesis 1. L intersects E in at least one point P, and 2. E does not contain L
a. 3. Indirect Proof Supposition- the intersection contains two points (Q also) a. i. ii. Which one of our hypothesis get refuted b/c of our drawing? Therefore, our supposition is false proving that theorem 3-2 is true C) Prove theorem 3-3 with an Indirect Proof a. Theorem 3-3 i. Given a line and a point not on the line, there is exactly 1 plane containing both of them 1. 2. We need to prove existence (at least one) and, a. i. Let Q and R be any two points of L. ii. By postulate 7 there is a plane E, containing P, Q, and R. iii. By Postulate 6, E contains L. Thus E contains P and L. b. prove uniqueness (at most one) a. supposition- suppose there are two planes containing the point and the line (the new plane is plane Y) i. Plane Y contains P and L. Then Plane Y contains P, Q and R ii. But P, Q and R are noncollinear b/c L is the only line that contains Q and R (Postulate 4), and L does not contain P. iii. Thus we have two different planes, E and Y containing the non-collinear points P, Q and R. This contradicts Postulate 7. Existence and uniqueness i. When we prove existence, we show that there is `at least one' ii. When we prove uniqueness, we show that there is `at most one' iii. "exactly one" or "one and only one" implies both existence and uniqueness
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This note was uploaded on 05/16/2011 for the course ALGEBRA 098 taught by Professor Johnson during the Spring '09 term at Grand Valley State University.
- Spring '09