If two straight lines meeting one another be parallel to two straight lines meeting one another, not being in the same plane, the planes through them are parallel.
如果两条相交直线分别平行于另一平面内的两条相交直线，且这两组直线不在同一平面内，那么通过这两组直线的平面互相平行。
For let the two straight lines AB, BC meeting one another be parallel to the two straight lines DE, EF meeting one another, not being in the same plane; I say that the planes produced through AB, BC and DE, EF will not meet one another. For let BG be drawn from the point B perpendicular to the plane through DE, EF [XI. 11], and let it meet the plane at the point G; through G let GH be drawn parallel to ED, and GK parallel to EF. [I. 31] Now, since BG is at right angles to the plane through DE, EF, therefore it will also make right angles with all the straight lines which meet it and are in the plane through DE, EF.
设直线AB、BC相交于B，分别平行于直线DE、EF相交于E，且两组直线不在同一平面内。
[XI. Def. 3] But each of the straight lines GH, GK meets it and is in the plane through DE, EF; therefore each of the angles BGH, BGK is right. And, since BA is parallel to GH, [XI. 9] therefore the angles GBA, BGH are equal to two right angles.
从B作BG垂直于平面通过DE、EF，垂足为G；过G作GH平行于ED，GK平行于EF。
[I. 29] But the angle BGH is right; therefore the angle GBA is also right; therefore GB is at right angles to BA. For the same reason GB is also at right angles to BC.
由于BG垂直于平面，故BG垂直于GH和GK，因此角BGH和BGK为直角。又因BA平行于GH，故角GBA等于角BGH，从而角GBA为直角，同理角GBC为直角。
Since then the straight line GB is set up at right angles to the two straight lines BA, BC which cut one another, therefore GB is also at right angles to the plane through BA, BC. [XI. 4] But planes to which the same straight line is at right angles are parallel; [XI. 14] therefore the plane through AB, BC is parallel to the plane through DE, EF.
因此BG垂直于BA和BC，从而垂直于平面通过AB、BC。而同一垂线垂直于两个平面，则两平面平行，故平面通过AB、BC平行于平面通过DE、EF。