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2-5 Postulates andParagraph

Proofs

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What is a Postulate?

A Postulate or axiom is a statement that is accepted as fact.

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Postulates: Points, Lines, and Planes

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Intersections of Lines and Planes

Postulate 2.6Postulate 2.6

If two lineslines intersect, then they intersect in exactly one pointpoint.

Postulate 2.7Postulate 2.7

If two planesplanes intersect, then they intersect in exactly one lineline.

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Example 1Identify the postulate that belongs to each statement.

Line m contains points F and G. Point E can also be on line m.

Lines s and t intersect at point D.

Points A, B, and C are noncollinear. Points A, B, and C determine a plane.

Planes P and Q intersect in line m

A line contains at least two points

If two lines intersect, then their intersection is a point.

Three noncollinear points determine a plane

The intersection of two planes is a line.

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Analyze Statements Using Postulates

Determine whether each statement is always, sometimes, or never true.

Points A and B are collinear.

Line r contains only point P.

Four points are noncollinear

If two coplanar lines intersect, then the point of intersection lies in the same plane as the two lines.

Always

Never

Sometimes

Always

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A postulate is a statement of “fact”

A Theorem is a statement that has been “proven” true

Proof: a logical argument in which each step towards the conclusion is supported by a definition, postulate, or theorem i.e. supporting statements.Paragraph Proof: a type of proof that gives the steps and supporting statements in paragraph formInformal Proof: paragraph proofs are a type of informal proof

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Theorem 2-1

Midpoint Theorem

If M is the midpoint of AB, then AM = MB

A BM

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Proofs

We will take a closer look at proofs in the next section.

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Have a great day!!