(5) All the straight lines perpendicular to a given line at a given (7) At a point in a plane but one straight line can be drawn perpen- (8) From a point without a plane only one perpendicular to the plane can be drawn. § 409. (9) Two straight lines perpendicular to the same plane are parallel. § 405. (10) If one of two parallel lines is perpendicular to a plane, the other is also. § 407. (11) Two planes perpendicular to the same straight line are parallel. § 412. (12) A straight line perpendicular to one of two parallel planes is also perpendicular to the other. § 422. (13) If a straight line is perpendicular to a given plane, every plane containing that line is perpendicular to the given plane. § 436. 7. THEOREMS ON STRAIGHT LINES AND PLANES PARALLEL TO THEM. (1) If two straight lines are parallel, any plane containing one of them, and not the other, is parallel to the other. § 413. (2) Through either of two given straight lines not lying in the same plane, one plane can be passed parallel to the other line. § 414. (3) Through a given point a plane can be passed parallel to any two given straight lines in space. § 415. (4) If a straight line is parallel to a given plane, it is parallel to the intersection of any plane through it, with the given plane. § 416. (5) If a straight line is parallel to a given plane, a line drawn from any point in the plane parallel to the given line lies in the given plane. § 417. (6) If two intersecting straight lines are each parallel to a given plane, the plane determined by these lines is also parallel to the given plane. § 418. 8. THEOREMS ON PLANES PERPENDICULAR TO EACH OTHER. (1) If a straight line is perpendicular to a given plane, every plane containing that line is perpendicular to the given plane. § 436. (2) Any plane perpendicular to the edge of a dihedral angle is perpendicular to each of its faces. § 438. (3) If two planes are perpendicular to each other, a straight line drawn in one of them, perpendicular to their intersection, is perpendicular to the other. § 439. (4) If two planes are perpendicular to each other, a straight line drawn from any point of their intersection, perpendicular to one plane, must lie in the other. § 440. (5) If two planes are perpendicular to each other, a straight line drawn from any point in one of them, perpendicular to the other, must lie in the first plane. § 442. (6) If two intersecting planes are each perpendicular to a third plane, their intersection is also perpendicular to that plane. § 443. 9. THEOREMS ON PARALLEL PLANES. (1) Two parallel planes are intersected by any third plane in parallel lines. § 419. (2) Parallel line-segments terminated by parallel planes are equal. $ 420. (3) Two parallel planes are everywhere equidistant. § 421. (4) A straight line perpendicular to one of two parallel planes is also perpendicular to the other. § 422. (5) If two straight lines are cut by three parallel planes, the corresponding segments are proportional. § 425. 10. THEOREMS ON DIHEDRAL ANGLES. (1) All plane angles of the same dihedral angle are equal. § 431. (3) Two dihedral angles are equal if their plane angles are equal. § 433. (4) Two dihedral angles are in the same ratio as their plane angles. § 434. (5) The locus of points equidistant from the boundaries of a dihedral angle is the plane bisecting that angle. § 444. 11. THEOREMS ON TRIHEDRAL AND POLYHEDRAL ANGLES. (1) The sum of any two face angles of a trihedral angle is greater than the third face angle. § 458. (2) Any face angle of a polyhedral angle is less than the sum of the remaining face angles. § 459. (3) The sum of the face angles of any convex polyhedral angle is less than four right angles. § 460. (4) If two trihedral angles have the three face angles of one equal, respectively, to the three face angles of the other, their corresponding dihedral angles are also equal. § 461. (5) An isosceles trihedral angle and its symmetrical trihedral angle are identically equal. § 463. (6) If two face angles of a trihedral angle are equal, the dihedral angles opposite them are also equal. § 464. (7) If two isosceles trihedral angles have the three face angles of one equal, respectively, to the three face angles of the other, they are identically equal. § 465. (8) If two trihedral angles have two face angles and the included dihedral angle of one equal, respectively, to two face angles and the included dihedral angle of the other, they are either identically equal or symmetrical, according as the parts are arranged in the same order or in opposite orders. § 466. (9) If two trihedral angles have a face angle and the two adjacent dihedral angles of one equal, respectively, to a face angle and the two adjacent dihedral angles of the other, they are either identically equal or symmetrical, according as the parts are arranged in the same order or in opposite orders. § 467. 12. MISCELLANEOUS THEOREMS. (1) Two straight lines each parallel to a third line are parallel to each other. § 396. (2) If from the foot of a given perpendicular to a plane, a straight line is drawn at right angles to any line of the plane, any line through their intersection which meets the given perpendicular is at right angles to the line of the plane. § 406. (3) Of all straight lines which can be drawn from a point to a plane, the perpendicular is the shortest. § 410. (4) If two intersecting straight lines lying in one plane are parallel, respectively, to two intersecting straight lines lying in another plane, the two planes must be parallel, and the angles formed by the lines are equal. § 424. (5) The acute angle which a straight line makes with its own projection upon a plane is the least angle it makes with any line of that plane. § 449. (6) Only one straight line can be drawn perpendicular to each of two given straight lines not lying in the same plane. § 451. (7) The common perpendicular to two given straight lines not lying in the same plane is the shortest line between them. § 452. CHAPTER VII PRISMS AND PYRAMIDS SECTION I AREA AND VOLUME OF A PRISM DEFINITIONS 468. A section of a surface made by an intersecting plane is the locus of points common to the surface and the plane. 469. A surface, such that every section of it made by an intersecting plane consists of one or more closed lines, is called a closed surface. 470. A closed surface which is made up wholly of intersecting planes is called a polyhedron. The planes are called the faces of the polyhedron; the lines in which the faces intersect are called the edges; the points in which the edges intersect are called the vertices. The line-segment connecting any two vertices not lying in the same face is called a diagonal of the polyhedron. If a polyhedron lies wholly on one side or the other of each of its faces, it is called a convex polyhedron. Any section of a convex polyhedron made by a plane is a convex polygon. A polyhedron of four faces is called a tetrahedron; one of five faces is called a pentahedron; one of six faces is called a hexahedron; one of eight faces is called an octahedron; one of twelve faces is called a dodecahedron; one of twenty faces is called an icosahedron. 471. A polyhedron of which two faces are convex polygons lying in parallel planes and identically equal, while the remaining faces are parallelograms, is called a prism. The equal parallel faces are called the bases of the prism; the remaining faces, the lateral faces. The edges lying in the bases are called the base edges; and the intersections of the lateral faces, the lateral edges. The base edges of a prism are equal and parallel, two and two, each edge of one base being equal and parallel to an edge of the other base. The lateral edges of a prism are all equal and parallel (Art. 420), and make equal angles with the plane of either base. A prism is called triangular, quadrilateral, etc., according as its bases are triangles, quadrilaterals, etc. 472. A right prism is one whose lateral edges are perpendicular to its bases. If the lateral edges are not perpendicular to the bases, the prism is called oblique. 473. A right section of a prism is the section made by any plane perpendicular to the lateral edges. 474. The lateral area of a prism is the sum of the areas of the lateral faces of the prism. 475. The altitude of a prism is the perpendicular distance between its bases. |