PPOB. XXVIII. THEOR. If a st. line falling upon two other st. lines makes the exterior angle equal to the interior opposite angle upon the same side of the line; or makes the int. angles upon the same side together equal to two angles; the two st. lines shall be parallel. DEM. P. 15. Ax. 1. P. 27. 13. Ax. 3. PROP. XXIX. THEOR. If a line fall upon two parallel st. lines, it makes the alternate angles equal to one another; the ext. angle equal to the int. and opp. angle upon the same side; and likewise the two int. angles upon PROP. XXX. THEOR. St. lines which are parallel to the same st. line are parallel to each other. DEM. P. 29. Ax. 1. P. 27. EXP. 1| Hyp. 2 Concl. CON. 1 Sup. DEM 1 C. Hyp. 2 P. 29. 3 C. Hyp. 4 P. 29. 5 D. 2. 6 Ax. 1. 7 P. 27. 8 Recap. PROP. XXXI. PROB. To draw a st. line through a given point, parallel to a given st. line. PROP. XXXII. THEOR. If a side of any triangle be produced, the ext. angle is equal to the two int. a angles; and the three int. angles of every triangle are together equal to two rt. angles. CON. P. 31. DEM. P. 29. Ax. 2. P. 13. Ax. 1. COR. I. All the int. angles of any rectilineal figure, together with four rt. angles, are equal to twice as angles as the figure has sides. DEM. P. 15. C. 2. |