Page:Carroll - Euclid and His Modern Rivals.djvu/296

258 . II. 3. Through a given point, without a given Line, a Line may be drawn such that the two Lines are equally inclined to any transversal.

Take a second point, on the same side of the given Line and at the same distance from it; and join the 2 points.

Then the Line, so drawn, and the given Line, are equally inclined to any transversal.

Therefore through a given point, &c.

. II. 18 (b).

The angles of a Triangle are together equal to two right angles.



Let ABC be a Triangle. It is to be proved that its 3 angles are together equal to 2 right angles.

Through A let DAE be drawn, such that DAE, BC are equally inclined to any transversal.

Then ∠B = ∠DAB, and ∠C = ∠EAC;

∴ ∠s B, C, BAC = ∠s DAB, EAC, BAC;

= 2 rt ∠s.

Therefore the angles &c.

. II. 4.

A Pair of Lines, which are equally inclined to a certain transversal, are so to any transversal.