PROPOSITION 31.

Through a given point to draw a straight line parallel to a given straight line.

Let A be the given point, and BC the given straight line;

thus it is required to draw through the point A a straight line parallel to the straight line BC.

Euclid's Elements, Book I, Proposition 31 Drawing a straight line EAF through a given point A parallel to a given straight line BC by joining AD and making angle EAD equal to angle ADC. E A F B D C

Let a point D be taken at random on BC, and let AD be joined; on the straight line DA, and at the point A on it, let the angle DAE be constructed equal to the angle ADC [I. 23];

and let the straight line AF be produced in a straight line with EA.

Then, since the straight line AD falling on the two straight lines BC, EF has made the alternate angles EAD, ADC equal to one another,

[I. 27]therefore EAF is parallel to BC.

Therefore through the given point A the straight line EAF has been drawn parallel to the given straight line BC.

See Commentary on Proposition I.31.