PLANE GEOMETRY. 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 For since BA has the same direction as ED, and BC the same as EF, the differ ence of direction of BA and BC must be E -D 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. A B C D G E Produce DE and FE so as to form the angle GEH; then (11) and proposition; therefore angle Bangle E. PARALLEL LINES. 13. Definition. Parallel Lines are such as have the same direction; as A B and CD. C -D 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. they cannot be approaching in either direction, that is, they For if they never meet 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, C H F, 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 AGH and G H C, or B G H and GHD. 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 BG 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 GHC, 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 E F 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 di rection (12). E G A C B H D F 2d. The alternate internal angles, AGH and GHD, or BG H and G H C, are equal, since their sides have opposite directions (12). 3d. PLANE GEOMETRY. The internal angles on the same side, AGH and G H C, or BGH and G HD, are supplements of each other; for AGH is the supplement of AGE (8), which has just been proved equal to GHC. In the same way be proved that BGH it and GHD 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. 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 EG B =GHD, or AGH = GHD, or BGH and GHD supplements of each other; then AB is parallel to C' D. For, if through the point G a line 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. 27. An Isosceles Triangle is one which has two of its sides equal; as D E F. 28. An Equilateral Triangle is one whose sides are all equal'; as IG H. PLANE GEOMETRY. 29. A Right Triangle is one which has a right angle; as JK L. The side opposite the right angle is called the hypothenuse. 30. An Obtuse-angled Triangle is one which has an obtuse angle; as MNO. M4 N K 31. An Acute-angled Triangle is one whose angles are all acute; as DEF. Acute and obtuse-angled triangles are called oblique-angled triangles. 32. The side upon which any polygon is supposed to stand is generally called its base; but in an isosceles triangle, as DEF, in which DE EF, the third side D F is always considered the base. = THEOREM VII. 33. The sum of the angles of a triangle is equal to two right angles. Let A B C be a triangle; the sum of its three angles, A, B, C, is equal to two right angles. Produce A C, and draw CD par allel to A B; then DCE— A, be- A ing external internal angles (17); C BCD B, being alternate internal angles (17); hence but DCE+BCD+BCA=A+B+BCA DCE + B C D + BCA = two right angles (7) ; therefore 34. Cor. 1. If two angles of a triangle are known, the third can be found by subtracting their sum from two right angles. |