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 E . def. 10. right 2 angle; but, if not, from the point B draw BE at right $ 31. 1. angles b to CD; therefore the angles CBE, EBD are two right anglese; and because CBE is equal to the two angles CBA, ABE together, add the angle EBD to each of these equals; therefore 02 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, there fore 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 di AX. the same are equal d 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 riglat angles. Wherefore, when a straight line, &c. Q. E. D. PROP. XIV. THEOR. IF, at a point in a straight line viher straight lines, upon the opposite sides of it, inake the adjacent angles together equal to two right angles, these two. straight lines shall be in one and the same straight line. in the same straight line with it; therefore, because the straight Book I. line AB makes angles with the straight line CBE, upon one sides of it, the angles ABC, ABE are together equal a to two' right & 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 CĎA, ABD : take away the con mon angle ABC, the remaining angle ABE is equal to the remaining angle b 3. 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 witn it but BD, which therefore is in the same straight line with CB. Wherefore, if at a point, &c. Q. E.D. 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. Because the straight line AE makes with CD the angles CEA, C AED, these angles are together equal 1 to two right angles. & 13. 1. Again, because the straight line A B D CEA, AED are equal to the angles AED, DEB. Take away the common angle AED, and the remaining angle CEA is equal b to the remaining angle DI B. In the same manner it can b 3. Ax. be demonstrated that the angles CEB, AED are equal. Therefore, if two straight lines, &c. Q. E. D. Cor. l. Froin 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 thåt all the angles made by any number of lines meeting in one point, are together equal to four right angles. Book 1 PROP. XVI. THEOR. a 10. 1. 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. A Because AE is equal to B C G D b 15. 1. 4. 1. d 15. 1. PROP. XVII. THEOR. ANY two angles of a triangle are together less than А Produce BC to D; and be. than the interior and opposite angle ABC ; to each of B C a 16. 1. these add the angle ACB; therefore the angles ACD, ACB Book I. 2.9 THE greater side of every triangle is opposite to the greater angle. D Let ABC be a triangle, of which А Because AC is greater than AB, a 3. 1. BD; and because ADB is the exterior angle of the triangle BDC; B с it is greater b than the interior and opposite angle DCB ; but b 16. 1. ADB is equal c 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 Let ABC be a triangle, of which the angle ABC is greater A 2 5. 1. Cb 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 PROP. XX. THEOR. Soe Nate. ANY two sides of a triangle are together greater than the third side. a 3. 1. F b 5. 1. Let ABC be a triangle ; any two sides of it together are D Because DA is equal to AC, the с E. D. PROP. XXI. THEOR. See Note, 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 |