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)*.