Proof. 1. Suppose X taken on OA so that OX = OY. 2. Then ▲ POX A POY, and PX = PY. 3. But PX, and .. its equal PY, > PA. For the converse: 4. Same as steps 1 and 2. Case 3 I, th. 20 5. .. PA < PX, ... PA < PY. Given 6. X cannot fall on A, for then PA= PX. 7. Nor between O and A, for then PA > PX. Why? Proof. 1. Suppose X taken on OA so that OX OY. 2. Then A POX A POY, and PX PY. I, th. 1 3. And ... OA <OY, or OX, ... PA< PX, or PY. Why? For the converse: 4. Same as steps 1, 2. 5. Same as steps 6, 7, 8 under converse of 4. 6. .. OA < OX, or its equal OY. DEFINITION. The length of the perpendicular from a point to a plane is called the distance from that point to the plane. Theorem 12. The acute angle which a line makes with its own projection on a plane is the least angle which it makes with any line in that plane. AA' drawn. jection of A, and OX made equal to OA', and AX, 2. Then AA' < AX. 3... in Th. 11, 1 OXA and OA'A, ZA'OA <<XOA. I, th. 11 Theorem 13. Parallel lines intersecting the same plane are equally inclined to it. DEFINITION. Two straight lines, AO, P'A', not coplanar, are regarded as forming an angle which is equal to the angle P'A'O' formed by P'A' and a line A'O', parallel to AO, drawn from a point A' on P'A'. See figure to th. 13. EXERCISE. 585. Parallel line-segments are proportional to their projections on a plane. Theorem 14. If a line intersects a plane, the line in the plane perpendicular to the projection of the first line at the point of intersection, is perpendicular to the line itself. Given AB intersecting the plane MN at A, B' the projection of B on B N D E EXC 2. Then ▲ AB'E A AB'E', M Why? DEFINITION. A line is said to be parallel to a plane when it never meets the plane, however far produced. In that case, also, the plane is said to be parallel to the line. Theorem 15. Any plane containing only one of two parallel lines is parallel to the other. Given the parallel lines AB, A'B', and the plane MN contain ing AB but not A'B'. B Р N 2. If A'B' meets MN it meets AB, ... AB and A'B' lie wholly in P. 3. But AB II A'B', this is impossible. Post. 7 Def. Il lines EXERCISE. 586. A line which is parallel to a plane is parallel to its projection on that plane. Theorem 16. Between two lines not in the same plane, one, and only one, common perpendicular can be drawn. Given two lines, K, L, not coplanar. Ο 3. Then K is not I to L', for then it would be I to L. 4. Let K intersect L' at P. Th. 4 Def. projection 5. Then . L' is the locus of the feet of all from 7... if PQ is drawn to both L and K. COROLLARY. to L, it is, and the only 1, The common perpendicular is the shortest line segment between two lines not in the same plane. For if Q'P' QP, then QPQ'P'<Q'R. Th. 11, 1. DEFINITION. The length of the common perpendicular from one line to another is called the distance between those lines. EXERCISES. 587. To construct a plane containing a given line, and parallel to another given line. (The construction assumed in step 1 of th. 16.) 588. A line parallel to each of two intersecting planes is parallel to their line of intersection. 589. If one of two parallel lines is parallel to a plane, what may be said of the other? Prove it. DEFINITIONS. Any number of planes containing the same line are said to form a pencil of planes; the line is called its axis. Any two planes of a pencil are said to form a dihedral angle. The two planes of a dihedral angle are called the faces, and the axis of the pencil is called the edge of the dihedral angle. Two intersecting planes form more than one dihedral angle, just as two intersecting lines form more than one plane angle, this term now being used to designate an angle made by lines in a plane. A plane of a pencil turning about the axis from one face of a dihedral angle to the other is said to turn through the angle, the angle being greater as the amount of turning is greater. Since the size of a dihedral angle depends only upon the amount of turning just mentioned, it is independent of the extent of the faces. If perpendiculars are erected from any point in the edge of a dihedral angle, one in each face, the size of the plane angle thus formed evidently varies as the size of the dihedral angle. Hence a dihedral angle is said to be measured by that plane angle, numerical measure. or, strictly, to have the same |