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

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258
APPENDIX III.

(κ). 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. [(θ).

Therefore through a given point, &c. Q. E. D.

(λ). 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. [(κ).

Then ∠B = ∠DAB, and ∠C = ∠EAC;

∴ ∠s B, C, BAC = ∠s DAB, EAC, BAC;

= 2 rt ∠s. [Euc. I. 13.

Therefore the angles &c. Q. E. D.

(μ). II. 4.

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