577 The acute angle which a straight line makes with its projection upon a plane is the least angle which it makes with any line in the plane. M B N HYPOTHESIS. BC is the projection of AB upon the plane MN, and BD is any other line in MN. 1164 If a line is parallel to a plane, it is parallel to its projection upon the plane. 1165 If a line is equal to its projection upon a plane, it is parallel to the plane. 1166 If two parallel lines intersect a plane, they make equal angles with the plane. 1167 A line and its projection upon a plane determine a second plane perpendicular to the first. PROPOSITION XXIV. THEOREM 578 Only one common perpendicular can be drawn between two straight lines not in the same plane. HYPOTHESIS. AB and CD are two lines not in the same plane. CONCLUSION. Only one common perpendicular can be drawn between AB and CD. PROOF Through CD pass a plane MN | to AB (§ 538), and through AB pass a plane AFL to MN, intersecting MN in EF. Since EF is to AB (§ 540), EF is not to CD (§ 536); hence EF and CD intersect in some point E. § 564 impos Draw GH to EF. Then GH is L to MN, We thus have GK and GH each to MN, which is sible. .. only one can be drawn between AB and CD. § 527 Q. E. D. 579 COROLLARY. The common perpendicular is the shortest line that can be drawn between two lines. POLYEDRAL ANGLES DEFINITIONS 580 A polyedral angle is the figure formed by three or more planes proceeding from a point. Thus, S-ABCD is a polyedral angle. The common point S is the vertex; the planes SAB, SBC, etc., are the faces; the intersections of the faces, SA, SB, etc., are the edges; and the angles ASB, BSC, etc., are the face angles of the polyedral angle. Two consecutive faces of a polyedral angle form a diedral angle. A S 581 The parts of a polyedral angle are its face angles and its diedral angles. 582 A polyedral angle is convex if a section made by a plane cutting all its faces is a convex polygon. 583 A triedral angle is a polyedral angle of three faces; a tetraedral angle is a polyedral angle of four faces, etc. 584 An isosceles triedral angle is a triedral angle having two equal face angles. 585 Triedral angles are rectangular, bi-rectangular, or trirectangular, according as they have one, two, or three right diedral angles. 586 Two polyedral angles are equal if their face and diedral angles are equal respectively and arranged in the same order. 587 Two polyedral angles are symmetrical if their face and diedral angles are equal respectively and arranged in reverse order; as, S-ABC and S'-A'B'C'. Symmetrical polyedral angles cannot, in general, be made to coincide. For example, the right glove will not fit the left hand. 588 Vertical polyedral angles are those which have a common vertex, and in which the edges of one are the prolongations of the edges of the other. PROPOSITION XXV. THEOREM 589 Vertical polyedral angles are symmetrical. HYPOTHESIS. S-ABCD and S-A'B'C'D' are vertical polyedral angles. PROOF The face angles ASB, BSC, etc., are equal to the face angles A'SB', B'SC', etc., respectively. § 112 The diedral angles SB and SB', being formed by the same two intersecting planes (§ 516), are vertical diedral angles and therefore equal. § 560 Likewise the other diedral angles are equal respectively. order. .. the polyedral angles are symmetrical. § 587 Q. E. D. PROPOSITION XXVI. THEOREM 590 The sum of any two face angles of a triedral angle is greater than the third. B HYPOTHESIS. In the triedral angle S-ABC, let the face angle ASC be greater than either of the face angles ASB or BSC. CONCLUSION. ZASB+2 BSC > < ASC. PROOF In the face ASC draw SD making ASD = ASB. = Take SD SB, and through B and D pass a plane cutting the triedral angle in the section ABC. But BC + AB > AC (2). Subtracting (1) from (2), BC > DC. In the A BSC and DSC, we have SB SD by const., SC common, and BC > DC. .: < BSC > < DSC. § 162 § 166 $.160 Ax. 5 Proved § 171 EXERCISE 1168. Find the locus of a point equidistant from two given planes, and also equidistant from two given points. |