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379. A polyedral angle of three faces is called a Tetraedral angle, one of four faces a Quadraedral, etc.

380. A triedral angle is called Isosceles if it has two of its face-angles equal; and Equilateral if three of its face-angles are equal.

381. Triedral angles are Rectangular, Bi-rectangular, or Trirectangular, according as they have one, two, or three right diedral angles.

382. A polyedral angle is Convex, if the polygon formed by the intersections of a plane with all its faces be a convex polygon.

383. Opposite or Vertical polyedral angles are those in which the edges of the one are prolongations of the edges of the other.

Such angles are symmetrical, as O-ABC and O-A'B'C'.

D

A

C

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PROPOSITION XII. THEOREM.

384. The sum of any two face-angles of a triedral angle is greater than the third.

A'

B

If the angles are equal, it is evident that the sum of two will be greater than the third.

If unequal, let OAC be greater than ZAOB or Z BOC in the triedral ≤ 0-ABC.

In the plane AOC draw the line OD making AOD= ZAOB; draw AC cutting OD in D and pass a plane through AC so that it may cut off OB equal to OD.

Then the triangles OAD and OAB will be equal (by 86), having two sides and the included angle equal by construction, which gives AD = AB.

B

or

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In the triangle ABC, AB+ BC > AC (by 6); subtracting the equals AB = AD, we have BC> DC.

In the triangles BOC and DOC, OB = OD, and the side OC is common, but the third side BC is greater than DC, therefore (by 95) BOC > < DOC.

Add the equal angles, ZAOB = LAOD,

and

ZAOB+≤ BOC >≤ AOD+≤ DOC,

ZAOB+Z BOC > ZAOC.

Q.E.D.

EXERCISES.

1. If two face-angles of a triedral angle are equal, the diedral angles opposite them are also equal.

2. The planes bisecting the diedral angles of a triedral angle intersect in a straight line.

3. The perpendicular bisectors of the faces of a triedral angle intersect in a straight line.

PROPOSITION XIII.

THEOREM.

385. The sum of the face-angles of any complex polyedral angle is less than four right angles.

D

B

Let O-ABCDE be a convex polyedral angle.

To prove that the sum of the face angles AOB, BOC, etc., is less than four right angles.

Pass the plane ABCDE intersecting the edges in A, B, C, D, and E, and let O' be any point in this plane.

Join O' with A, B, C, D, and E.

Since the sum of any two face-angles at a triedral angle is greater than the other two (by 384),

ZOAB+ 2 OAE > < EAB,

< OBA+≤ OBC > < ABC, etc.

also

That is, the sum of the base angles whose vertex is O is greater than the sum of the base angles whose vertex is O'.

But the sum of all the angles of the triangles whose vertex is O must be equal to the sum of all the angles of the triangles whose vertex is O', since the number of triangles in each case is the same, and (by 79) the value of the angles of each triangle is identical.

Therefore the angles at the vertex of the triangles, having the common vertex O, is less than the vertex angles at O', or less than four right angles.

Q.E.D.

BOOK VII.

POLYEDRONS, CYLINDERS, AND CONES.

GENERAL DEFINITIONS.

386. A Polyedron is a solid bounded by planes. The Faces are the bounding planes, the Edges are the intersections of its faces, and the Vertices are the intersections of its edges.

387. The Diagonal of a polyedron is a straight line joining any two non-adjacent vertices not in the same plane.

388. A polyedron of four faces is called a Tetraedron; of six faces, a Hexaedron; of eight faces, an Octaedron; of twelve faces, a Dodecaedron; of twenty faces, an Icosaedron.

389. A polyedron is called Convex when the section made by any plane is a convex polygon.

All polyedrons treated hereafter will be understood to be

convex.

390. The Volume of a solid is the number which expresses its ratio to some other solid taken as a unit of volume. The Unit of Volume is a cube whose edge is a linear unit.

391. Two solids are Equivalent when their volumes are equal.

PRISMS AND PARALLELOPIPEDS.

392. A Prism is a polyedron two of whose faces are equal and parallel polygons, and the other faces are parallelograms.

The equal and parallel polygons are called the Bases of the prism; the parallelograms are the Lateral Faces; the lateral faces taken together form the Lateral or Convex Surface; and the intersections of the lateral faces are the Lateral Edges.

The lateral edges are parallel and equal, and the area of the lateral surface is called the Lateral Area.

393. The Altitude of a prism is the perpendicular distance between its bases.

394. Prisms are Triangular, Quadrangular, Pentangular, etc., according as their bases are triangles, quadrangles, pentagons,

etc.

395. A Right Prism is a prism whose lateral edges are perpendicular to its bases.

396. An Oblique Prism is a prism whose lateral edges are oblique to its bases.

397. A Regular Prism is a right prism whose bases are regular polygons, and hence its lateral faces are equal rectangles.

RIGHT PRISM.

398. A Truncated Prism is a portion of a prism included between either base and a section inclined to the base and cutting all the lateral edges.

399. A Right Section of a prism is a section perpendicular to its lateral edges.

400. A Parallelopiped is a prism whose bases are parallelograms; therefore all the faces are parallelograms, and the opposite faces are equal and parallel.

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