that have their sides which are terminated in one extremity of the base equal to one another, and likewise their sides terminated in the other extremity: But this is impossible (1.7.); therefore, if the base BC coincides with the base EF; the sides BA, AC cannot but coincide with the sides ED, DF; wherefore likewise the angle BAC coincides with the angle EDF, and is equal (Ax. 8.) to it. Therefore if two triangles, &c. Q. E. D. PROP. IX. PROB. To bisect a given rectilineal angle, that is, to divide it into two equal angles. Let BAC be the given rectilineal angle, it is required to bisect it. Take any point D in AB, and from AC cut (1. 3.) off AE equal to AD; join DE, and upon it, on the side towards B and C, describe (1. 1.) an equilateral triangle DEF; then join AF; the straight line AF bisects the triangle BA C. A Because AD is equal to AE, and AF is common to the two triangles DAF, EAF; the two sides DA, AF, are equal to the two sides EA, A F, each to each; and the base DF is equal to the base EF; therefore the angle DAF is equal (1. 8.) to the angle EAF; wherefore the given rectilineal angle BAC is bisected by the B straight line AF. Which was to be done. D PROP. X. PROB. To bisect a given finite straight line, that is, to divide it into two equal parts. Let AB be the given straight line; it is required to divide it into two equal parts. Describe (i. 1.) upon it an equilateral triangle ABC, and bisect (1. 9.) the angle ACB by the straight line CD. cut into two equal parts in the point D. AB is с Because AC is equal to CB, and CD common to the two triangles ACD, BCD; the two sides AC, CD are equal to BC, CD, each to each; and the angle ACD is equal to the angle BCD (Constr.); therefore the base AD is equal to the base (1. 4.) DB, and the straight line AB is divided into two equal parts in the point D. Which was to be done. A D B PROP. XI. PROB. To draw a straight line at right angles to a given straight line, from a given point in the same. Let AB be a given straight line, and c a point given in it; it is required to draw a straight line from the point cat right angles to a B. F Take any point D in AD, and (1. 3.) make CE equal to CD, and upon DE describe (1. 1.) the equilateral triangle DFE, and join FC; the straight line FC drawn from the given point c is at right angles to the given straight line AB. A D с E B Because DC is equal to CE, and FC common to the two triangles DCF, ECF; the two sides DC, CF, are equal to the two EC, CF, each to each; and the base DF is equal to the base EF; therefore the angle DCF is equal (1. 8.) to the angle ECF; and they are adjacent angles: but, when the adjacent angles which one straight line makes with another straight line are equal to one another, each of them is called a right (1. Def. 10.) angle; therefore each of the angles DCF, ECF, is a right angle. Wherefore, from the given point c, in the given straight line AB, FC has been drawn at right angles to AB. Which was to be done. COR. By help of this problem, it may be demonstrated that two straight lines cannot have a common segment. E If it be possible, let the two straight lines ABC, ABD have the segment AB common to both of them. From the point B draw BE at right angles to AB; and because ABC is a straight line, the angle CBE is equal (1. Def. 10.) to the angle EBA; in the same manner, because ABD is a straight line, the angle DBE is equal to the angle EBA; wherefore the angle DBE is equal to the angle CBE, the less to the greater, A which is impossible; therefore two straight lines cannot have a common segment. B D PROP. XII. PROB. To draw a straight line perpendicular to a given straight line of an unlimited length, from a given point without it. Let AB be the given straight line, which may be produced to any length both ways, and let c be a point without it. It is required to draw a straight line perpendicular to AB from the point c. C A Take any point D upon the other side of AB, and from the centre C, at the distance CD, describe (Post. 3.) the circle EGF meeting AB in F, G; and bisect (1. 10.) FG in H, and join CF, CH, CG; the straight line CH, drawn from the given point c, is perpendicular to the given straight line AB. D Because FH is equal to HG, and HC common to the two triangles FHC, GHC, the two sides FH, HC are equal to the two GH, HC, each to each; and the base CF is equal (1. Def. 15.) to the base CG; therefore the angle CHF is equal (1. 8.) to the angle CHG; and they are adjacent angles; but when a straight line standing on a straight line makes the adjacent angles equal to one another, each of them is a right angle, and the straight line which stands upon the other is called a perpendicular to it (Def. 10.); therefore from the given point c a perpendicular CH has been drawn to the given straight line AB. Which was to be done. GB E PROP. XIII. THEOR. The angles which one straight line makes with another upon the one side of it, are either two right angles, or are together equal to two right angles. 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. A E A For, if the angle CBA be equal to ABD, each of them is a right (Def. 10.) angle; but, if not, from the point в draw BE at right angles (1. 11.) to CD; therefore the angles CBE, EBD are two right angles (Def. 10.); 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 equal (Ax. D в с D B с 2.) 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 (Ax. 1.) 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. PROP. XIV. THEOR. If, 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 A B, make the adjacent angles ABC, ABD, equal together to two right angles. BD is in the same straight line with CB. E с B D For, if BD be not in the same straight line with CB, let BE be in the same straight line with it; therefore, because the straight line A B makes angles with the straight line CBE, upon one side of it, the angles ABC, ABE are together equal (1. 13.) to two right angles; but the angles ABC, ABD are likewise together equal to two right angles (Hyp.); therefore the angles CBA, ABE are equal to the angles CBA, ABD. Take away the common angle ABC, the remaining angle ABE is equal (Ax. 3.) to the remaining angle 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.E.D. PROP. XV. THEOR. If two straight lines cut one another, the vertical, or op posite, 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 EAD. Because the straight line AE makes C with CD the angles CEA, AED, these angles are together equal (1. 13.) to two A right angles. Again, because the straight line DE makes with AB the angles AED, DEB, these also are together equal (1. 13.) 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 (Ax. 3.) to the remaining angle DEB. 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. B 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 (1. 10.) AC in E, join BE and produce it to F, and make EF equal to BE; join also FC, and produce AC to G. A F B 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 (1. 15.) to the angle CEF, because they are opposite vertical angles; therefore the base AB is equal (1. 4.) to the base CF, and the triangle A EB 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 |