Two Planes have Line in Common

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.

$\blacksquare$

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Sources

• 1952: T. Ewan Faulkner: Projective Geometry (2nd ed.) ... (previous) ... (next): Chapter $1$: Introduction: The Propositions of Incidence: $1.2$: The projective method: The propositions of incidence