let BE be in the same straight line with it; therefore, be- Book I. it may be demonstrated, that no other can be in the same straight line with it but BD, which therefore is in the same straight line with CB. Wherefore, if at a point, &c. Q. E. D. PROP. XV. THEOR. IF two straight lines cut one another, the vertical, 1 or opposite, angles shall be equal. Let the two straight lines AB, CD cut one another in the For the angles CEA, B a 13. 1. E 'D Therefore, if two straight lines, &c. Cor. 1. From this it is manifest, that, if two straight lines COR. 2. And hence, all the angles made by any number of straight lines, meeting in one point, are together equal to four right angles. Book 1. Monely PROP. XVI. THEOR. IF one side of a triangle be produced, the exterior angle is greater than either of the interior and oppo. site angles. Let ABC be a triangle, and let its side BC be produced to D; the exterior angle ACD is greater than either of the interior opposite angles CBA, А. F Bisecta ĄC in E, join BE F, and E Because AE is equal to EC, D each; and the angle AEB is equal to the - angle CEF, because they are vertical angles; therefore the base AB is equal c to the base CF, G and the triangle AEB to the triangle CEF, and the remaining angles to the remaining angles, each to each, to which the equal sides are opposite; wherefore the angle BAE is equal to the angle ECF; but the angle ECD is greater than the angle ECF; therefore the angle ECD, that is, ACD, is greater than BAE. In the same manner, if the side BC Be bisected, it may be demonstrated that the angle BCG, that is the angle ACD, is greater than the angle ABC. Therefore, if one side, &c. Q. E. D. b 15. 1. C 4. 1. PROP. XVII. THEOR. ANY two angles of a triangle are together less than two right angles, Let ABC be any triangle ; Book I. A Produce BC to D; and a 16. 1. interior and opposite angle ABC; to each of these add the angle ACB; therefore B C D the angles ACD, ACB are greater than the angles ABC, ACB; but ACD, ACB are together equal to two right b 13. 1. angles; therefore the angles ABC, BCA are less than two right angles. In like manner, it may be demonstrated, that BAC, ACB, as also CAB, ABC, are less than two right angles. Therefore, any two angles, &c. Q. E. D. THE greater side of every triangle has the greater angle opposite to it. Let ABC be a triangle, of A D Ċ a 3. 1. caus Book I. PROP. XIX. THEOR. ang gre BC THE greater angle of every triangle is subtended by the greater side, or has the greater side opposite ani iti tha sid to it. be a 5. 1. Let ABC be a triangle, of which the angle ABC is greater than the angle BCA; the side AC is likewise greater than the side AB. For, if it be not greater, AC B b 18. 1. PROP. XX. THEOR. ANY-two sides of a triangle are together greater than the third side. a 3. 1. Let ABC be a triangle ; any two sides of it together are Produce BA to the point Because DA is equal to C b 5. 1. cause the angle BCD of the triangle DCB is greater than its Book I. angle BDC, and that the greater side is opposite to the greater angle: therefore the side DB is greater than the side c 19. 1. BC; but DB is equal to BA and AC together; therefore BA and AC together are greater than BC. In the same manner it may be demonstrated, that the sides AB, BC are greater than CA, and BC, CA greater than AB. Therefore any two sides, &c. Q. E. D. PROP. XXI. THEOR. IF from the ends of one side of a triangle there N. be drawn 'two straight lines to a point within the triangle, these two lines shall be less than the other two sides of the triangle, but shall contain a greater angle. Let the two straight lines BD, CD be drawn from B, C, the ends of the side BC of the triangle ABC, to the point D within it; BD and DC are less than the other two sides BA, AC of the triangle, but contain an angle BDC greater than the angle BAC. Produce BD to E ; 'and because two sides of a trianglea are a 20. 1. greater than the third side, the two sides BA, AE of the triangle ABE are greater than BE. To each of these add EC; therefore the sides BA, AC are А. greater than BE, EC. Again, because the two sides CE, ED, of the triangle CED, are greater than CD, add DB to each of these; therefore the sides CE, EB are greater than CD, DB; but it has been shown that BA, AC are greater than BE, EC; much more then are BA, AC greater than BD, DC. B C Again, because the exterior angle of a triangleb is greater b 16.1. than the interior and opposite angle, the exterior angle BDC of the triangle CDE is greater than CED; for the same reason, the exterior angle CEB of the triangle ABE is greater than BAC; and it has been demonstrated that the angle |