# Two Planes have Line in Common

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## Theorem

Two distinct planes have exactly one (straight) line in common.

## Proof

Take two distinct lines in plane $1$.

From Propositions of Incidence: Plane and Line, they each meet plane $2$ in one point each, say at $A$ and $B$.

Thus $A$ and $B$ both lie in both planes.

Thus the line defined by $A$ and $B$ lies in both planes.

$\blacksquare$

## Sources

- 1952: T. Ewan Faulkner:
*Projective Geometry*(2nd ed.) ... (previous) ... (next): $1.2$: The projective method