Descriptive geometry . rob. 12, Special Case 6, § 118), and thesecondary projection Axv passes through sx and the secondaryprojection, b^, of b. In other respects the constructions ofFigs. 242 and 243 follow that of Fig. 241. Special Case I. The line of intersection of the givenplanes is parallel to H or V Let Q and R, Fig. 244, be thegiven planes, with HQ and HR parallel. Then the line of in-tersection, A, is parallel to H (Prob. 12, Special Case 1,§ 118). Pass the auxiliary plane Z perpendicular to A. Since XVI, § 146] PROBLEMS ON THE LINE AND PLANE 161 A is parallel to H, the plane Z is per
Descriptive geometry . rob. 12, Special Case 6, § 118), and thesecondary projection Axv passes through sx and the secondaryprojection, b^, of b. In other respects the constructions ofFigs. 242 and 243 follow that of Fig. 241. Special Case I. The line of intersection of the givenplanes is parallel to H or V Let Q and R, Fig. 244, be thegiven planes, with HQ and HR parallel. Then the line of in-tersection, A, is parallel to H (Prob. 12, Special Case 1,§ 118). Pass the auxiliary plane Z perpendicular to A. Since XVI, § 146] PROBLEMS ON THE LINE AND PLANE 161 A is parallel to H, the plane Z is perpendicular to H. Theline A intersects the plane Z in the point c (cA, c) (Prob. 13,Special Case, § 119). Revive the plane Z about HZ into revolved position, cr, is obtained simply by making thedistance from HZ to cr equal to the distance of C from it is evident that the lines of intersection of Z with thegiven planes Q and R will take the positions crs2 and crs3, and cv Av \t>^^ > -. > ,s>. Fig. 244. Fig. 245. the included angle, 0, between these lines is the angle betweenthe given planes. Note. The student may discover that TrZ, and also c, are superfluousin this construction. Special Case II. One of the given planes is parallel toII or V. This case falls under the solution just given ; a stillsimpler solutionis shown in Fig. 245. The given plane R isparallel to II. Therefore the angle between planes Qand B isequal to the angle, 6, which Q makes with //, the latter anglebeing found by Problem 18, § 136. 146. Application of the Preceding Problem. As instances inwhich the angle between two planes is wanted in actual con-struction, we may cite the case of two intersecting pitch (slop-ing) roofs, or of two intersecting masonry walls with battered(sloping) faces. The way in which the problem usually appearsis as a corollary to the following problem. M 162 DESCRIPTIVE GEOMETRY [XVI, § 146 Problem 28. Given the horizontal traces of two planes, and thea
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