straight line, it will be perpendicular to that line at its middle point. 78. If two triangles have two sides of the one equal to two sides of the other, each to each, and the included angle of the one greater than the included angle of the other, the third side of that which has the greater angle will be greater than the third side of the other. Of the two sides DE, D F, let D F be the side which is not shorter than the other; make the angle EDG equal to BAC; and make D G equal to AC or D F, and join EG, GF. Since D F, or its equal D G, is not shorter than DE, it is longer than DH (Prop. XIV. Cor. 3); therefore its extremity, F, must fall below the line EG. The two triangles, ABC and DE G, have the two sides AB, AC equal to the two sides DE, DG, each to each, and the A included angle BAC of the one equal to the included angle EDG of the other; hence the side BC is equal to EG (Prop. V. Cor.). . In the triangle DFG, since DG is equal to DF, the angle D F G is equal to the angle D G F (Prop. VII.); but the angle DGF is greater than the angle EGF; therefore the angle DFG is greater than EGF, and much more is the angle EFG greater than the angle EGF. Because the angle EFG in the triangle EFG is greater than EGF, and because the greater side is opposite the greater angle (Prop. X.), the side EG is greater than EF; and EG has been shown to be equal to BC; hence BC is greater than EF. PROPOSITION XVII.-THEOREM. 79. If two triangles have two sides of the one equal to two sides of the other, each to each, but the third side of the one greater than the third side of the other, the angle contained by the sides of that which has the greater third side will be greater than the angle contained by the sides of the other. Let ABC, DEF be two triangles, the side AB equal to D E, and AC equal to DF, and A D the side CB greater than EF, then will the angle B C E F For, if it be not greater, it must either be equal to it or less. But the angle A cannot be equal to D, for then the side BC would be equal to E F (Prop. V. Cor.), which is contrary to the hypothesis; neither can it be less, for then the side BC would be less than EF (Prop. XVI.), which also is contrary to the hypothesis; therefore the angle A is not less than the angle D, and it has been shown that is not equal to it; hence the angle A must be greater than the angle D. PROPOSITION XVIII.-THEOREM. 80. If two triangles have the three sides of the one equal to the three sides of the other, each to each, the triangles themselves will be equal. E, and the angle C to the angle F, and the two triangles will also be equal. For, if the angle A were greater than the angle D, since the sides A B, A C are equal to the sides D E, DF, each to each, the side BC would be greater than EF (Prop. XVI.); and if the angle A were less than D, it would follow that the side BC would be less than EF. But by hypothesis BC is equal to EF; hence the angle A can neither be greater nor less than D; therefore it must be equal to it. In the same manner, it may be shown that the angle B is equal to E, and the angle C to F; hence the two triangles must be equal. 81. Scholium. In two triangles equal to each other, the equal angles are opposite the equal sides; thus the equal angles A and D are opposite the equal sides B C and EF. PROPOSITION XIX.-THEOREM. 82. If two right-angled triangles have the hypothenuse and a side of the one equal to the hypothenuse and a side of the other, each to each, the triangles are equal. The two triangles are evidently equal, if the sides BC and EF are equal (Prop. XVIII.). If it be possible, let ALTERNATE INTERIOR ANGLES lie within the parallels, and on different sides of the secant line, but are not adjacent to each other; as the angles BGH, GHC, and also AGH, GHD. ALTERNATE EXTERIOR ANGLES lie without the parallels, and on different sides of the secant line, but not adjacent to each other; as the angles EGB, CHF, and also the angles AGE, DHF. OPPOSITE EXTERIOR and INTERIOR ANGLES lie on the same side of the secant line, the one without and the other within the parallels, but not adjacent to each other; as the angles EGB, GHD, and also EGA, GHC, are, respectively, the opposite exterior and interior angles. PLANE FIGURES. 19. A PLANE FIGURE is a plane terminated on all sides by straight lines or curves. The boundary of any figure is called its perimeter. 20. When the boundary lines are straight, the space they enclose is E called a RECTILINEAL FIGURE, or D POLYGON; as the figure ABCDE. A B 21. A polygon of three sides is called a TRIANGLE; one of four sides, a QUADRILATERAL; one of five, a PENTAGON; one of six, a HEXAGON; one of seven, a HEPTAGON; one of eight, an OCTAGON; one of nine, a NONAGON; one of ten, a DECAGON; one of eleven, an UNDECAGON; one of twelve, a DODECAGON; and so on. The side opposite to the right angle is called the hypothenuse; as the side J L. 24. An ACUTE-ANGLED TRIANGLE is one which has three acute angles; as the triangles ABC and DEF, Art. 22. An OBTUSE-ANGLED TRIANGLE is one which has an obtuse angle; as the triangle G H I, Art. 22. Acute-angled and obtuse-angled triangles are also called oblique-angled triangles. 25. A PARALLELOGRAM is a quadrilateral which has its opposite sides parallel. 26. A RECTANGLE is any parallelogram whose angles are right angles; as the parallelogram A B C D. D C A B |