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A postulate is a statement that is assumed true without proof. A theorem is a true statement that can be proven. Listed below are six postulates and the theorems that can be proven from these postulates.

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Postulate 1: A line contains at least two points. Postulate 2: A plane contains at least three noncollinear points. Postulate 3: Through any two points, there is exactly one line. Postulate 4: Through any three noncollinear points, there is exactly one plane. Postulate 5: If two points lie in a plane, then the line joining them lies in that plane. Postulate 6: If two planes intersect, then their intersection is a line. Theorem 1: If two lines intersect, then they intersect in exactly one point. Theorem 2: If a point lies outside a line, then exactly one plane contains both the line and the point. Theorem 3: If two lines intersect, then exactly one plane contains both lines.

Example 1: State the postulate or theorem you would use to justify the statement made about each figure.

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Figure 1Illustrations of Postulates 1–6 and Theorems 1–3.

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(a)

Through any three noncollinear points, there is exactly one plane (Postulate 4).

(b)

Through any two points, there is exactly one line (Postulate 3).

(c)

If two points lie in a plane, then the line joining them lies in that plane (Postulate 5).

(d)

If two planes intersect, then their intersection is a line (Postulate 6).

(e)

A line contains at least two points (Postulate 1).

(f)

If two lines intersect, then exactly one plane contains both lines (Theorem 3).

(g)

If a point lies outside a line, then exactly one plane contains both the line and the point (Theorem 2).

(h)

If two lines intersect, then they intersect in exactly one point (Theorem 1).


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