Page:The Elements of Euclid for the Use of Schools and Colleges - 1872.djvu/255

Rh straight line GK at right angles to EF. [I. 11.

Then, because EF is at right angles to GH and GK, [Construction. EF is at right angles to the plane HGK passing through them. [XI. 4. And EF is parallel to AB; [Hypothesis. therefore AB is at right angles to the plane HGK. [XI. 8. For the same reason CD is at right angles to the plane HGK. Therefore AB and CD are both at right angles to the plane HGK. Therefore AB is parallel to CD. [XI. 6.

Wherefore, if two straight lines &c.

PROPOSITION 10. THEOREM.

If two straight lines meeting one another he parallel to two others that meet one another, and are not in the same plane with the first two, the first two and the other two shall contain equal angles.

Let the two straight lines AB, BC, which meet one another, be parallel to the two straight lines DE, EF, which meet one another, and are not in the same plane with AB, BC: the angle ABC shall be equal to the angle DEF.

Take BA, BC, ED, EF equal to one another, and join AD, BE, CF, AC, DF.

Then, because AB is equal and parallel to DE, therefore AD is equal and parallel to BE. [1. 33. For the same reason, CF is equal and parallel to BE. Therefore AD and CF are each of them equal and parallel to BE. Therefore AD is parallel to CF, [XI. 9