Book I. 2. Def. 10. 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 right b. 11. 1. angle. but if not, from the point B draw BE at right angles b to CD. therefore the angles CBE, EBD are two right anglesa. and because CBE is equal to the two angles CBA, ABE together; add the angle EBD to each of these equals, therefore the angles CBE, EBD are e. 2. Ax. 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 demonftrated to be equal to the same three angles; and things d. 1. Ax. 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. I Fat PROP. XIV. THEOR. at a point in a straight line, two other straight 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 fame straight line with CB. For if BD be not in the fame C straight line with CB, let BE be A. E B D in the same straight line with it. therefore because the straight line Book I. AB makes angles with the straight line CBE, uport one fide of it, the angles ABC, ABE are together equal to two right angles; but the a. 13. 1. angles ABC, ABD are likewife 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 ABD, the less to the greater, which b. 3. Ax. 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. E. Da IF PROP. XV. THEOR. F two straight lines cut one another, the vertical, oppofite, angles shall be equal. or Let the two ftraight lines AB, CD cut one another in the point E. the angle AEC shall be equal to the angle DEB, and CEB to AED. Because the straight line AE makes with CD the angles CEA, AED, these angles åre together equal to two right angles. again, 2. 13. 1. CEA, AED are equal to the angles AED, DEB. take away the common angle AED, and the remaining angle CEA is equal bb. 3. Α to the remaining angle DEB. In the fame manner it can be demonftrated 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 confequently that all the angles made by any number of lines meeting in one point, are together equal to four right angles. B PROP. Book I. 2. 10. 1. b. 15. 1. PROP. XVI. THEOR. IF one fide of 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 op posite angles CBА, ВАС. Bisect AC in E, joìm BE and produce it to F, and make because they are opposite ver tical angles. therefore the G C. 4. I. base AB is equal to the C d. rs. 1. bafe CF, and the triangle AEB to the triangle CEF, and the remaining angles to the remaining angles, each to each, to which the equal fides 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 ACD is greater than BAE. in the fame manner, if the fide BC be bisected, it may be demonstrated that the angle BCG, that is, the angle ACD, is greater than the angle 4. 15. 1. ABC. therefore if one side, &c. Q. E. D. A PROP. XVII. THEOR. NY two angles of a triangle are together less than two right angles. Let ABC be any triangle; any two of its angles together are lefs than two right angles. Produce BC to D; and because ACD is the exterior angle of the triangle ABC, ACD is greater than the interior and opposite angle ABC; to each of a B A 19 these add the angle ACB, therefore the angles ACD, ACB are great- Book 1. er than the angles ABC, ACB. but ACD, ACB are together equalb to two right angles; therefore the angles ABC, BCA are less than b. 13. 1. two right angles. in like manner it may be demonftrated that BAC, ACB, as alfo CAB, ABC are less than two right angles. therefore any two angles, &c. Q. E. D. T PROP. XVIII. THEOR. HE greater fide of every triangle is opposite Let ABC be a triangle of which the fide AC is greater than the side AB; the angle ABC is also greater than the angle BCA. B A tổ thế D 2. 3. 1, C Because AC is greater than AB, make AD equal to AB, and join BD. and because ADB is the exterior angle of the triangle BDC, it is greater than the interior and opposite angle b. 16. 1. DCB. but ADB is equal to ABD, because the side AB is e- c. s. i. qual to the fide AD; therefore the angle ABD is likewife greater than the angle ACB; wherefore much more is the angle ABC greater than ACB. therefore the greater fide, &c. Q. E. D. T PROP. XIX. THEOR. HE greater angle of every triangle is fubtended by Let ABC be a triangle of which the angle ABC is greater than the angle BCA. the fide AC is likewife greater than the fide AB. For if it be not greater, AC must either be equal to AB, or less then the angle ABC would beless A 2. 5. 1. C than Book I. b than the angle ACB; but it is not; therefore the side AC is not ☑ less than AB. and it has been shewn that it is not equal to AB. therefore AC is greater than AB. wherefore the greater angle, &c, Q. E. D. b 18.1. See N. A PROP. XX. THEOR. together greater than NY two fides of a triangle are Let ABC be a triangle; any two fides of it together are greater than the third side, viz. the fides BA, AC greater than the fide BC; and AB, BC greater than AC; and BC, CA greater than AB. c. 19. 1. er than the angle ADC. and because the angle BCD of the triangle DCB is greater than its angle BDC, and that the greater fide is opposite to the greater angle, therefore the fide DB is greater than the fide BC. but DB is equal to BA and AC; therefore the fides BA, AC are greater than BC. in the same manner it may be demonftrated that the fides AB, BC are greater than CA; and Bс, CA greater than AB. therefore any two fides, &c. Q. E. D. PROP. XXI. THEOR. See N. F from the ends of the side of a triangle there be drawn two ftraight 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 fides BA, AC of the triangle, but contain an angle BDC greater than the angle BAC. Produce BD to E; and because two fides of a triangle are greater than the third side, the two fides BA, AE of the triangle ABE are. |