Proposition 20. Theorem.* 550. The acute angle between a straight line and its projection on a plane is the least angle which the line makes with any line of the plane. Hyp: Let BC be the projection of the st. line AB on the plane MN, and М. let BD be any other st. line in the plane, drawn through B, the pt. in D which AB meets the plane. 'N B 1. All points whose projections upon a plane lie in a straight line are themselves in one plane: how is this plane defined ? 2. If AB, BC, CD are lines, such that ABC, BCD are right angles, and AB is at right angles to the plane BCD, then will CD be at right angles to the plane ABC. Proposition 21. Theorem.* 551. Between two straight lines not in the same plane one, and only one, common perpendicular can be drawn. Hyp. Let AB and CD be the given st. lines. A G B To prove there is one I, and no M more, to both AB and CD. Proof. Through one of the lines, AD K as CD, pass a plane MN || to the Hlor D line AB. (514) Through AB pass the plane AB' I to MN, and let their intersection A'B' meet CD at A'. Then A'B' is | to AB. (517) At A' erect A'A in the plane AB' I to A'B'. ... A'A is I to the plane MN, (53) and .:. A'A is I to the line CD. (487) Because A'A is I to A'B', .. it is I to AB. (71) .:. A'A is 1 to both the lines AB and CD. If there is any other common I to AB and CD, let it be HG. Because HG is I to AB, ... it is I to a line HL drawn || to AB in the plane (71) and ... HG is I to the plane MN. (500) But GK in the plane AB', I to A'B', is I to MN. (537) .. from the pt. G there are two Is GK, GH to MN. But this is impossible. (502) .:. HG is not a common I to the lines AB, CD. ... A'A is the only common I to AB and CD. Q.E.D. 552. Cor. The perpendicular A'A is the least distance between AB and CD; for any other line GH > GK (497), or its equal A'A. MN, . EXERCISES. 1. Parallel lines intersecting the same plane make equal angles with it. (492) 2. If a plane bisects a line perpendicularly, every point of the plane is equally distant from the extremities of the line. 3. If three lines in space are parallel, in how many planes may they lie when taken two at a time? 4. If four lines in space are parallel, in how many planes may they lie when taken two at a time? 5. How many different planes may the sides of a quadrilateral in space contain when taken two and two ? 6. If two lines not in the same plane are intersected by the same line, how many planes may be determined by the three lines taken two and two ? 7. A straight line makes equal angles with parallel planes. 8. The sum of two adjacent diedral angles, formed by one plane meeting another, is equal to two right diedral angles. 9. If two planes intersect each other, the opposite or vertical diedral angles are equal. 10. When a plane intersects two parallel planes, the alternate-interior diedral angles are equal, and the exteriorinterior diedral angles are equal. 11. Show that two observations with a spirit-level are sufficient to determine if a plane is horizontal: and prove that for this purpose the two positions of the level must not be parallel. 12. To draw a straight line perpendicular to a given plane from a given point outside of it. 13. To draw a straight line perpendicular to a given plane from a given point in the plane. POLYEDRAL ANGLES. DEFINITIONS. 553. When three or more planes meet in a common joint, they are said to form a polyedral angle at that point. The common point in which the planes meet is the vertex of the angle, the intersections of the planes are the edges, the portions of the planes between the edges $ are the faces, and the plane angles formed by the edges are the face-angles. Thus, the point S is the vertex, the straight lines SA, SB, etc., are the edges, the planes A SAB, SBC, etc., are the faces, and the angles ASB, BSC, etc., are the face-angles of the polyedral angle S-ABCD. 554. The edges of a polyedral angle may be produced indefinitely; but to represent the angle clearly, the edges and faces are supposed to be cut off by a plane, as in the figure above. The intersection of the faces with this plane forms a polygon, as ABCD, which is called the base of the polyedral angle. 555. In a polyedral angle, each pair of adjacent faces forms a diedral angle, and each pair of adjacent edges forms a face-angle. There are as many edges as faces, and therefore as many diedral angles as faces. 556. The magnitude of a polyedral angle depends only upon the relative position of its faces, and is independent of their extent. Thus, by the face SAB is not meant the triangle SAB, but the indefinite plane between the edges SA, SB produced indefinitely. 557. Two polyedral angles are equal, when the face and diedral angles of one are respectively equal to the face and diedral angles of the other, taken in the same order. 558. A polyedral angle of three faces is called a triedral angle; one of four faces is called a tetraedral angle; etc. 559. A polyedral angle is convex when its base is a convex polygon. (141) 560. A triedral angle is called isosceles when it has two of its face-angles equal; when it has all three of its faceangles equal it is called equilateral. 561. A triedral angle is called rectangular, bi-rectangular, or tri-rectangular, according as it has one, two, or three, right diedral angles. The corner of a cube is a tri-rectangular triedral angle. 562. Two polyedral angles are symmetrical, when the face and diedral angles of one are equal to the face and diedral angles of the other, each to each, but arranged in reverse order. Thus, the triedral angles S - ABC, S'-A'B'C' are symmetrical when the face-angles ' ASB, BSC, CSA are equal respectively to the face-angles A'S'B', B'S'C', C'S'A', and the diedral angles SA, SB, SC to the diedral angles S'A', S'B', S'C'. CA' When two polyedral angles are symmetrical, it is, in general, impossible to bring them into coincidence. The two hands are an illustration. The right hand is symmetrical to the left hand, but cannot be made to coincide with it. The right glove will not fit the left hand, but is symmetrical to it. 563. Opposite or vertical polyedral angles are those in which the edges of one are the prolongations of the edges of the other. AA |