Common Section of Two Planes is Straight Line

Proof

 * Euclid-XI-3.png

Let $AB$ and $BC$ be two distinct planes that cut one another.

Let $DB$ be their common section.

Suppose $DB$ were not a straight line.

Then let:
 * the straight line segment $DEB$ be drawn in the planes $AB$

and:
 * the straight line segment $DFB$ be drawn in the planes $BC$.

Thus the two straight line segments $DEB$ and $DFB$ have the same endpoints.

Thus $DEB$ and $DFB$ enclose an area, which is absurd.

Therefore $DEB$ and $DFB$ are not straight lines.

Hence the result.