PROP. XIII. THEOR. GEN. ENUN. The angles, which one straight line makes with another upon one side of it, are either two right angles, or are together equal to two right angles. PART. ENUN.-Let the st. with CD, upon one side of it, thes CBA, ABD; then these are either two rt. ≤s, or their sum two rt. s. If the CBA = ABD, line AB make as in fig. 1, then each of them D (fig. 2) draw BE at rt. Zs to the add to each the ▲ EBD; = ▲ CBA + ABE + 2. Again, since the add to each the Fig. 1. A. B Fig. 2. E A B CBE + EBD EBD (Ax. 2). DBA = DBE + EBA, ABC; .'. DBA + ABC = ▲ DBE + EBA + ABC (Ax. 2). 3. .. ≤ DBA + ABC = CBE + EBD (AX. 1) = two rt. s. Wherefore the < which one st. line, &c.— Q. E. D. It is clear from this Proposition that the sum of all the <", which are made by any number of st. lines drawn to the same pt. in another st. line, on the same side of it, two rt. 48. = The Proposition is of great practical use in Trigonometry and Astronomy. We are now in a position to furnish a very simple demonstration of the equality of the above and below the base of an isosc. A. Thus : PART. ENUN.-Let ABC be an isosc. ▲, having the side AB = the side AC; and let the sides AB, AC be produced to D and E; then the ABC = L ACB, and the DBC = LECB. CONST.-Bisect the BAC by the A L CAF;.* BAF = the ABC = Z above the base are =. E 2. Again, the 4s CBA + CBD = two rt. 4$ = 2s BCA + BCE (Prop.XIII.); but ▲ ABC ACB; ... remaining 4 DBC remaining ECB: i. e. s below the base are = -Q. E. D. == GEN. ENUN.—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. PART. ENUN.-At the pt. B in the st. line AB, let the two st. lines BC, BD make the adjacent ABC, ABD = two rt.s; then BC and BD are in the same st. line. A E -D B DEMONST. (Ad absurd.) : For, suppose that BD be not in the same st. line with BC, and that BE is in the same st. line with it. S 1. Then, because AB makes with CBE, upon one side of it, the s CBA, ABE; .. the s CBA + ABE = two rt. s. (Prop. XIII.) 2. But the $ CBA + ABD two rt. Zs. (Hyp.) 3... CBA + ABE = Ls СВА + ABD (Ax. 1). 4. . (by subtracting the common CBA) the ABE 2 ABD (Ax. 3): i. e. the < = >, which is impossible. 5. .. BE is not in the same st. line with BC. 6. In like manner it may be proved that no other can be in the same st. line with it but BD, which is .. in the same st. line with it. Wherefore, if at a pt. &c.-Q. E. D. This Proposition is the converse of the preceding. It is necessary that the two lines BC, BD should meet from opposite sides of AB; for it is very possible that on the same side two lines may form s to two rt. 4s with it, without being in the same st. line. Thus : = CONST.-Let c be any pt. in a st. line AB. Draw CD. at rt. 4s to Ab. (Prop. XI.) Bisect the ACD by the st. line cE. (Prop. IX.) In co take any pt. D, and through D draw DE at rt. 4s to CD. Produce ED to F, and make DF ED. (Prop. III.) Join CF. DEMONST.-Then in ▲ CDE, CDF, ED, DC = FD, DC, each to each, and the EDC rt. ▲ FDC; ... base cE rt. LFCD = half a rt. base CF (Prop. IV.); (Const.): ... the ACF + ACE a rt. + half a rt. 4, and the E = two rt. 4; and they are formed by two st. lines EC, FC, meeting AB in the same pt. The said lines, however, are not in the same st. line. GEN. ENUN.-If two straight lines cut one another, the vertical, or opposite, angles shall be equal. E B PART. ENUN.-Let the two st. lines AB, CD cut one another in the pt. c. E; then the AEC shall be = DEB, and the A CEB shall be = AED. DEMONST.-1. Because the st. line AE makes, with CD, the s CEA, AED; ..theses are = two rt. s. (Prop. XIII.) 2. Also, because the st. line DE makes, with AB, the AED, DEB; .. these two rt. s. (Prop. XIII.) are also 3... the 4 CEA + AED = < AED + deb (Ax. 1). 4... (by subtraction) the CEA = ≤ DEB. 5. In like manner it may be shown that CEBAED. Wherefore, if two st. lines, &c.—Q. E. D. COR. 1.-From this it is manifest that, if two straight lines cut one another, the < which they make at the pt. of intersection are together to four rt. Zs. COR. 2.-Consequently, the sum of all the Zs made by any number of st. lines intersecting at one pt. is to four rt. s. As the student should take nothing for granted, he will do well to demonstrate the equality of the 4 CEB and AED, as stated in the Proposition.-Though capable of a rigorous demonstration, the Proposition is nevertheless but little more than an inference from the definition of an 4. For if two straight lines, inclined at an ▲, intersect, their inclination must continue the same on both sides; or, in other words, the verticals will be =. With respect to the Corollaries, it is clear that CE makes, with AB, the $ AEC + two rt. 4, and DE makes, with AB, the 2 AED + DEB= two rt. 4 (Prop. XIII.); ... by addition, the ▲ AEC four rt. 2. Again, as these four / * are merely subdivided by any number of st. lines intersecting in E, the sum of four rt. ≤ remains unaltered. The practical application of the 15th Proposition to the measurement of 4 may be thus illustrated: CEB = + CEBAED + DEB Let ABCD be a graduated (Obs. on Def. 18), having an object-glass, or telescope, moveable round the cr. E; let the glass be directed to the object s, and the degree marked to which it points on the Cce; then let it be turned round till it points to another object T, and the degree marked as before. The 4 distance between the objects, or the number of degrees which the glass has passed over, is clearly that which is contained in the AEB, measured by the arc AB; or, which is the same thing, the number contained in the Z CED, measured by the arc CD. Upon this principle the Theodolite, and several astronomical instruments are constructed. |