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Unformatted text preview: 63 Theorems on Lines and Planes
A) Postulate 47 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 32 with an Indirect Proof a. Theorem 32 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 32 is true C) Prove theorem 33 with an Indirect Proof a. Theorem 33 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 noncollinear 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
 Johnson
 Algebra

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