Since AE A'E', AF-A'F', and Z BAC=Z B'A'C', (Above)

▲ AEF = ▲ A'E'F', and EF = E'F'.

(66)

(465)

.. dihedral Z AS = dihedral ▲ A's',

(being measured by equal ✩ FDE, F'D'E'.)

In the same way it may be proved that

dihed. BS dihed. ▲ B's', and dihed. / Cs = dihed. ≤ c's'. Hence the trihedrals S-ABC, S'-A'B'C', are equal or symmetrical according as the equal parts are or are not arranged in the same order. Q.E.D. (494)

496. COR. 1. If two trihedral angles have the three face angles of the one respectively equal to the three face angles of the other, the dihedral angles of the one are respectively equal to the dihedral angles of the other.

497. COR. 2. An isosceles trihedral angle, that is, one having two of its face angles equal, is equal to its symmetrical trihedral. For if in S-ABC, we have ASB = BSC, then in S'-A'B'C' we shall have ZA'S'B' = LB'S'C'; and the CSA, C'S'A' will have equal faces on each side of them; then also the dihedral angles will be similarly arranged, and the trihedrals will be equal (495).

EXERCISE 677. If a line makes an acute angle with a plane, every plane with which it makes the same angle is parallel to the first.

678. Parallel lines that intersect the same plane make equal angles with it.

679. If the projections of any number of points upon a plane lie in one straight line, these points are in one plane. What plane?

Geom. - 17

EXERCISES.

QUESTIONS.

680. What space concepts are determined in position by one point, by two, by three, respectively?

681. To what theorem in Book I. does Prop. V. correspond?

682. What is the locus of all the points in space that are equidistant from a given circumference ?

683. What is the locus of all the points in space that are equidistant from the vertices of a triangle ?

684. What is the locus of all the points in space that are equidistant from the mid points of the sides of a triangle ?

685. To what theorem in Book I. does Cor. 3 of Prop. VI. correspond?

686. What is the locus of all the lines that are perpendicular to a given line at a given point?

687. To what theorems in Book I. do Prop. VIII. and its first corollary correspond?

688. What is the locus of all the lines that have a given line in a given plane as their projection ?

689. What is the locus of all the points in space that are at a given distance from a given plane ?

690. To what theorems in Book I. do Prop. XII., Cor., and Prop. XIII. correspond?

691. To what theorem in Book IV. does Prop. XIV. correspond? 692. To what theorem in Book I. does Prop. XIX. correspond? 693. What is the locus of all the points in space that are equidistant from two intersecting planes ? From two parallel planes ?

THEOREMS.

694. If a line is perpendicular to one of two intersecting planes, its projection on the other plane is perpendicular to the intersection.

695. A plane parallel to two sides of a quadrilateral in space, - that is, a quadrilateral having its sides two and two in different planes, divides the other two sides proportionally.

696. The mid points of the sides of a quadrilateral in space are the angular points of a parallelogram.

697. If the intersections of any number of planes are parallel, the perpendiculars drawn to these planes from the same point in space are in the same plane.

698. If a line is equally inclined to both faces of a dihedral angle, the points in which it meets the faces are equally distant from the edge of the dihedral.

699. If a line makes equal angles with three lines in the same plane, it is perpendicular to that plane.

700. If a plane be passed through one diagonal of a parallelogram, the perpendiculars to that plane from the extremities of the other diagonal are equal.

701. If from a point within a dihedral angle, perpendiculars are drawn to its faces, the angle contained by these perpendiculars is equal to the plane angle of the adjacent dihedral angle formed by producing one of the planes.

702. If three planes have a common intersection, perpendiculars to these planes from any point in the intersection are in the same plane.

703. In any trihedral angle, the three planes bisecting its dihedral angles intersect in the same line.

704. In any trihedral angle, the three planes passed through its edges perpendicular to the opposite faces, intersect in the same line.

705. In any trihedral angle, the three planes passed through the edges and the bisectors of the opposite face angles, intersect in the same line.

706. In any trihedral angle, the three planes passed perpendicularly through the bisectors of the face angles, intersect in the same line.

LOCI.

Find the loci of the points in space that respectively satisfy the following conditions:

707. Are equidistant from two given points.

708. Are equidistant from two given intersecting lines.

709. Have their distances from two given planes in a given ratio.

710. Are equidistant from the vertices of a given triangle.

711. Are equidistant from the sides of a given triangle.

712. Are equidistant from the vertices of a quadrilateral whose opposite angles are supplementary.

713. Are equidistant from the circumference of a given circle.

714. Are equidistant from three given planes.

715. Are equidistant from the edges of a given trihedral angle.

716. Are equidistant from two given planes and two given points in space.

PROBLEMS.

In the construction of the following problems, it is assumed that, besides the constructions of Plane Geometry, we are able: (1) to pass a plane through any given line and any point or line that can be in the same plane with it (422, 423, 424); (2) through any given point in or without a given plane, to draw a perpendicular to that plane.

717. Through a given line in a plane pass a plane making a given angle with that plane.

718. Through a given line without a given plane pass a plane making a given angle with that plane.

719. Through the edge of a given dihedral angle pass a plane bisecting that angle.

720. Through a given point without a given plane pass a plane parallel to that plane.

721. At a given distance from a given plane pass a plane parallel to that plane.

722. Through a given point pass a plane perpendicular to a given straight line.

723. Through the vertex of a given trihedral angle draw a line making equal angles with the edges.

724. In a given plane find a point such that the lines drawn to it from two given points without the plane, shall make equal angles with the plane. (Two cases.)

725. In a given plane find a point equidistant from three given points without the plane.

726. In a given straight line find a point equidistant from two given points not in the same plane as the line.

BOOK VIII.

POLYHEDRONS.

498. A polyhedron is a solid bounded by four or more polygons. The bounding polygons are called the faces; their intersections, the edges; and the intersections of the edges, the vertices of the polyhedron.

The least number of planes that can form a solid angle the trihedral is three. As the faces and edges extend indefinitely, the space within the angle is of indefinite extent, so that in order to cut off a definite portion of that space, a fourth plane must be passed intersecting the faces. Hence four is the least number of planes that can inclose a space.

499. A polyhedron of four faces is called a tetrahedron; one of six faces, a hexahedron; of eight faces, an octahedron; of twelve faces, a dodecahedron; of twenty faces, an icosahedron.

500. A polyhedron is convex when every section of it by a plane is a convex polygon. As none but convex polyhedrons are to be treated of in what follows, the term polyhedron will always signify convex polyhedron.

501. The volume of a polyhedron is its quantity as measured by the polyhedron taken as unit of volume, or is the numerical measure of that quantity.

In every-day life, volume is expressed by stating how many stated solid measures a given solid contains; as 356 In abstract discussions, however, by volume is.