Book 1. Let the straight line AB make with CD, upon one side of it, the angles CBA, ABD; these are either two right angles, or are together equal to two right angles. For, if the angle CBA be equal to ABD, each of them is a A E A a def. 10. right angle; but, if not, from the point B draw BE at right b11. 1. angles to CD; therefore the angles CBE, EBD are two right angles; and because CBE is equal to the two angles CBA, ABE together, add the angle EBD to each of these equals; therefore 2 Ax. the angles CBE, EBD are equal to the three angles CBA, ABE, EBD. Again, because the angle DBA is equal to the two angles DBE, EBA, add to these equals the angle ABC; therefore the angles DBA, ABC are equal to the three angles DBE, EBA, ABC; but the angles CBE, EBD have been demonstrated to be equal to the same three angles; and things that are equal to the same are equal to one another; therefore the angles CBE, EBD are equal to the angles DBA, ABC; but CBE, EBD are two right angles; therefore DBA, ABC are together equal to two right angles. Wherefore, when a straight line, &c. Q. E. D. d1 Ax. d PROP. XIV. THEOR. her straight IF, at a point in a straight line lines, upon the opposite sides of it, make the adjacent angles together equal to two right angles, these two. straight lines shall be in one and the same straight line. At the point B in the straight line AB, let the two straight lines BC, BD upon the opposite sides of AB, make the adjacent angles ABC, ABD equal together to two right angles: BD is in the same straight line with CB. For, if BD be not in the same straight line with CB, let BE beC A E B in the same straight line with it; therefore, because the straight Book I. line AB makes angles with the straight line CBE, upon one side of it, the angles ABC, ABE are together equal to two right a 13.1. angles; but the angles ABC, ABD are likewise together equal to two right angles; therefore the angles CBA, ABE are equal to the angles CBA, ABD : take away the common angle ABC, the remaining angle ABE is equal to the remaining angle b S. Ax ABD, the less to the greater, which is impossible; therefore BE is not in the same straight line with BC. And, in like manner, 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. ED. PROP. XV. THEOR. ́ IF two straight lines cut one another, the vertical or opposite, angles shall be equal. Let the two straight lines AB, CD cut one another in the point E; the angle AEC shall be equal to the angle DEB, and CEB to AED. a 13. 1. E B D Because the straight line AE makes with CD the angles CEA, C AED, these angles are together equala to two right angles. Again, because the straight line A DE makes with AB the angles AED, DEB, these also are to gether equal to two right angles; and CEA, AED have been demonstrated to be equal to two right angles; wherefore the angles CEA, AED are equal to the angles AED, DEB. Take away the common angle AED, and the remaining angle CEA is equal to the remaining angle DB. In the same manner it can be demonstrated that the angles CEB, AED are equal. Therefore, if two straight lines, &c. Q. E. D. COR. 1. From this it is manifest, that, if two straight lines cut one another, the angles they make at the point where they cut, are together equal to four right angles. COR. 2. And consequently that all the angles made by any number of lines meeting in one point, are together equal to four right angles. D b 3. Ax. Book 1. 10. 1. b 15. 1. 4. 1. PROP. XVI. THEOR. IF one side of a triangle be produced, the exterior angle is greater than either of the interior opposite 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, BAC. Bisect AC in E, join BE and produce it to F, and make EF equal to BE; join also FC, and produce AC to G. Because AE is equal to EC, and BE to EF; AE, EB are equal to CE, EF, each to each; and the angle AEB is equal to the angle CEF, because they are opposite vertical angles; therefore the base AB is equal to the base CF, and the triangle A 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 LCF; therefore the angle ACD is greater than BAE: in the same manner, if the site BC be bisected, it may be demonstrated that the angle d 15. 1. BCG, that is, the angle ACD, is greater than the angle ABC. Therefore, if one side, &c. Q. E. D. a 16. 1. PROP. XVII. THEOR. ANY two angles of a triangle are together less than two right angles. Let ABC be any triangle; any two of its angles together are less than two right angles. Produce BC to D; and because ACD is the exterior angle of the triangle ABC, ACD is greater a than the interior and opposite angle ABC; to each of B A these add the angle ACB; therefore the angles ACD, ACB Book I. are greater than the angles ABC, ACB; but ACD, ACB ~ are together equal to two right angles; therefore the angles b 13. 1. 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. PROP. XVIII, THEOR. 2.9 THE greater side of every triangle is opposite to the greater angle. Let ABC be a triangle, of which the side AC is greater than the side AB; the angle ABC, is also greater than the angle BCA. Because AC is greater than AB, make AD equal to AB, and join BD; and because ADB is the ex terior angle of the triangle BDC, B c A Ꭰ a 3. 1. C it is greater than the interior and opposite angle DCB; but b 16. 1. ADB is equal to ABD, because the side AB is equal to the c 5. 1. side AD; therefore the angle ABD is likewise greater than the angle ACB; wherefore much more is the angle ABC greater than ACB. Therefore the greater side, &c. Q. E. D. PROP. XIX. THEOR. The greater angle of every triangle is subtended by the greater side, or has the greater side opposite to it. 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 must either be equal to AB, or less than it; it is not equal, because then the angle ABC would be equal to the angle ACB; but it is not; therefore AC is not equal to AB; neither is it less; because then the angle ABC, would be less than the angle ACB; B A a 5. 1. Съ 18. 1 Book I, but it is not; therefore the side AC not less than AB; and it has been shown that it is not equal to AB; therefore AC is greater than AB. Wherefore the greater angle, &c. Q. E. D. See No a 3. 1. b 5. 1. c 19. 1. See Note. PROP. XX. THEOR. ANY two sides of a triangle are together greater than the third side. Let ABC be a triangle'; any two sides of it together are greater than the third side, viz. the sides BA, AC greater than the side BC; and AB, BC greater than AC; and BC, GA greater than AB. Produce BA to the point D, and make AD equal to AC; and join DC. to Because DA is equal to AC, the c D C angle ADC; and because the angle BCD of the triangle DCB is greater than its angle BDC, and that the greater side is opposite to the greater angle; therefore the side DB is greater than the side BC; but DB is equal to BA and AC; therefore the sides BA, AC are greater than BC. In the same manner it may be demonstrated that the sides AB, BC are greater than CA, and BC, CA are greater than AB. Therefore any two sides, &c. E. D. PROP. XXI. THEOR. IF, from the ends of the side of a triangle, there be drawn two straight lines to a point within the triangle, these 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 triangle are greater than the third side, the two sides BA, AE of the tri |