to two angles (7)*; therefore the sum of all the angles at the point B is equal to four right angles. THEOREM II. 10. If at a point in a straight line two other straight lines upon opposite sides of it make the sum of the adjacent angles equal to two right angles, these two lines form a straight line. Let the straight line D B meet the two lines, AB, BC, so as to make ABD DBC= two right angles: then AB and BC form a straight line. A B D E C For if A B and B C do not form a straight line, draw BE so that A B and B E shall form a straight line; then the part equal to the whole, which is absurd (Axiom 6); therefore A B and B C form a straight line. THEOREM III. 11. If two straight lines cut each other, the vertical angles are equal. Let the two lines, AB, CD, cut cach other at E; then For AED is the supplement of both A D E * The figures alone refer to an article in the same Book; in referring to an article in another Book the number of the Book is prefixed. THEOREM IV. 12. Two angles whose sides have the same or opposite directions are equal. 1st. Let BA and BC, including the angle B, have respectively the same direction as ED and EF, including the angle E; then angle Bangle E. B A F D For since BA has the same direction as ED, and BC the same as EF, the difference of direction of BA and BC must be the same as the difference of direction of E D and E F; that is, angle Bangle E. 2d. Let BA and B C, including the angle B, have respectively opposite directions to ED and E F, including the angle E; then angle Bangle E. Produce DE and FE so as to form the angle GEH; then (11) GEH= DEF E 13. Definition. Parallel Lines are such as have the same direction; as A B and CD. 14. Corollary. Parallel lines can never meet. For, since parallel lines have the same direction, if they coincided at one point, they would coincide throughout and form one and the same straight line. Conversely, straight lines in the same plane that never meet, however far produced, are parallel. For if they never meet they cannot be approaching in either direction, that is, they must have the same direction. 15. Axiom. Two lines parallel to a third are parallel to each other. E A 16. Definition. When parallel lines are cut by a third, the angles without the parallels are called external; those within, internal; thus, AGE, EG B, CHF, FHD are external angles; AGH, BGH, GHC, GHD are internal angles. Two internal angles on the same side of the C G B H D F secant, or cutting line, are called internal angles on the same side; as A G H and G H C, or B G H and G HD. Two internal angles on opposite sides of the secant, and not adjacent, are called alternate internal angles; as A G H and G HD, or B G H and GH C. Two angles, one external, one internal, on the same side of the secant, and not adjacent, are called opposite external and internal angles; as EGA and G H C, or E G B and G H D. THEOREM V. 17. If a straight line cut two parallel lines, 1st. The opposite external and internal angles are equal. 2d. The alternate internal angles are equal. 3d. The internal angles on the same side are supplements of each other. Let EF cut the two parallels A B and CD; then 1st. The opposite external and internal angles, EGA and GHC, or EGB and G HD, are equal, since their sides have the same direction (12). E G B A H D F C or 2d. The alternate internal angles, A G H and GHD, BGI and GHC, are equal, since their sides have opposite directions (12). 3d. The internal angles on the same side, AGH and GHC, or B G H and G H D, are supplements of each other; for AGH is the supplement of AGE (8), which has just been proved equal to GIIC. In the same way it may be proved that BGH and G HD are supplements of each other. THEOREM VI. CONVERSE OF THEOREM V. 18. If a straight line cut two other straight lines in the same plane, these two lines are parallel, 1st. If the opposite external and internal angles are equal. 2d. If the alternate internal angles are equal. 3d. If the internal angles on the same side are supplements of each other. Let EF cut the two lines A B and CD so as to make E G B = G II D, or AGH = GHD, or BGH and· GHD supplements of each other; then A B is parallel to CD. For, if through the point G a line E B A D C F is drawn parallel to CD, it will make the opposite external and internal angles equal, and the alternate internal angles equal, and the internal angles on the same side equal (17); therefore it must coincide with AB; that is, A B is parallel to CD. PLANE FIGURES. DEFINITIONS. 19. A Plane Figure is a portion of a plane bounded by lines either straight or curved. When the bounding lines are straight, the figure is a polygon, and the sum of the bounding lines is the perimeter. 20. An Equilateral Polygon is one whose sides are equal each to each. 21. An Equiangular Polygon is one whose angles are equal each to each. 22. Polygons whose sides are respectively equal are mutually equilateral. 23. Polygons whose angles are respectively equal are mutually equiangular. Two equal sides, or two equal angles, one in each polygon, similarly situated, are called homologous sides, or angles. 24. Equal Polygons are those which, being applied to each other, exactly coincide. 25. Of Polygons, the simplest has three sides, and is called a triangle; one of four sides is called a quadrilateral; one of five, a pentagon; one of six, a hexagon; one of eight, an octagon; one of ten, a decagon. 28. An Equilateral Triangle is one whose sides are all equal; as IG H. H |