## Postulates and Theorems

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.

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

• (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).