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postulate or axiom -- A postulate is a statement that is accepted as true without mathematical proof.
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Postulate 2.1 -- Through any 2 points there will be exactly one line.
(There will definitely be one and only one line…)
aka…. Two points determine a line.
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Postulate 2.2 -- Through any 3 noncollinear points there is exactly one plane.
aka ….. Three noncollinear points determine a plane.
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Postulate 2.3 -- A line contains at least 2 points
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Postulate 2.4 -- A plane contains at least 3 noncollinear points
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Postulate 2.5 -- If two points lie in a plane, then the entire line containing those points lie in the plane.
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Postulate 2.6 -- If two lines intersect, their intersection is a point.
Intersection….points that two
objects have in common.
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Postulate 2.7 -- If two planes intersect, their intersection is a line.
Example #1: State the postulate that can be used to show each statement is true.
a. Planes and
intersect in line r.
If two planes intersect their intersection is a line.
The intersection of two planes is a line.
b. Lines r and n intersect at point D.
If two lines intersect, they intersect in a point.
c. Plane contains the points A, F, & D.
If you have a plane, it must contain at least three noncollinear points.
Example #2: Determine whether each statement is always, sometimes or never true. Explain your reasoning.
a. The intersection of three planes is a line.
Sometimes –
Three sides of a cube do not intersect in the same line.
Also…see the diagrams at right.
.
b. Line r contains only one point P.
Never, every line contains at least two points.
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Proof -- A proof is a logical argument in which each statement is supported (justified) by a reason that is accepted
(known) to be true.
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theorem -- A theorem has been (mathematically) proven to be true.
deductive argument -- A deductive argument is a chain of statements linking the given to what you are trying to prove.
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paragraph proof -- A paragraph proof is written in paragraph form and explains why a conjecture is true.
informal proofs -- Informal proofs, such as paragraph proofs, are no less valid than a formal 2-column proof.
Thm 2.1 Midpoint Theorem -- If M is the midpoint of AB , then AM  MB
.
A
M
: L2.5 16 – 29, 33– 41
B