Also, the right angled triangles ANI and BGD are similar, since, by hypothesis, the angles AIN and BDG are equal. Hence,

AI: BD :: AN : BG.

Therefore, AK: BH :: AI : BD.

Thus, by the aid of similar triangles, it may be proved that any two homologous lines, in two similar tetraedrons, have the same ratio as two homologous edges.

635. Thecrem. Two tetraedrons are similar when their faces are respectively similar triangles, and are similarly arranged.

For we know, from the similarity of the triangles, that every line made on the surface of one may have its homologous line in the second, making angles equal to those made by the first line.

If lines be made through the figure, it may be shown, by the aid of auxiliary lines, as in the corresponding proposition of similar triangles, that every possible angle in the one figure has its homologous equal angle in the other.

The student may draw the diagrams, and go through the details of the demonstration.

636. If the similar faces were not arranged similarly, but in reverse order, the tetraedrons would be symmetrically similar.

637. Corollary. Two tetraedrons are similar when three faces of the one are respectively similar to those of the other, and they are similarly arranged. For the fourth faces, having their sides proportional, are similar also.

638. Corollary. Two tetraedrons are similar when two triedral vertices of the one are respectively equal to two of the other, and they are similarly arranged.

639. Corollary.-Two tetraedrons are similar when the edges of one are respectively proportional to those of the other, and they are similarly arranged.

640. Theorem. - The areas of homologous faces of similar tetraedrons are to each other as the squares of their edges.

This is only a corollary of the theorem that the areas of similar triangles are to each other as the squares of their sides.

641. Corollary. The areas of homologous faces of similar tetraedrons are to each other as the squares of any homologous lines.

642. Corollary. The area of any face of one tetraedron is to the area of a homologous face of a similar tetraedron, as the area of any other face of the first is to the area of the homologous face of the second.

643. Corollary.—The area of the entire surface of one tetraedron is to that of a similar tetraedron as the squares of homologous lines.

TETRAEDRONS CUT BY A PLANE.

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644. Theorem. If a plane cut a tetraedron parallel to the base, the tetraedron cut off is similar to the whole.

For each triangular side is cut by a line parallel to its base (572), thus making all the edges of the two tetraedrons respectively proportional.

645. Theorem.-If two tetraedrons, having the same altitude and their bases on the same plane, are cut by a plane parallel to their bases, the areas of the sections will have the same ratio as the areas of the bases.

If a piane parallel to the bases pass through the vertex A, it will also pass through the vertex B (622). But

such a plane is parallel to the cutting plane GHP (566).

Therefore, the tetraedrons AGHK and BLNP have equal altitudes.

The tetraedrons AEIO and AGHK are similar (644). Therefore, EIO, the base of the first, is to GHK, the base of the second, as the square of the altitude of the first is to the square of the altitude of the second (641). For a like reason, the base CDF is to the base LNP as the square of the greater altitude is to the square of the less.

Therefore, EIO GHK :: CDF: LNP.

By alternation,

EIO: CDF :: GHK : LNP.

646. Corollary. When the bases are equivalent the sections are equivalent.

647. Corollary.-When the bases are equal the sections are equal. For they are similar and equivalent.

REGULAR TETRAEDRON.

648. There is one form of the tetraedron which deserves particular notice. It has all its faces equilateral. This is called a regular tetraedron.

649. Corollary. It follows, from the definition, that

the faces are equal triangles, the vertices are of equal triedrals, and the edges are of equal diedral angles.

650. The area of the surface of a tetraedron is found by taking the sum of the areas of the four faces. When two or more of them are equal, the process is shortened by multiplication. But the discussion of this matter will be included in the subject of the areas of pyra

mids.

The investigation of the measures of volumes will be given in another connection.

EXERCISES.

651.-1. State other cases, when two tetraedrons are similar, in addition to those given, Articles 635 to 639.

2. In any tetraedron, the lines which join the centers of the opposite edges bisect each other.

3. If one of the vertices of a tetraedron is a trirectangular triedral, the square of the area of the opposite face is equal to the sum of the squares of the areas of the other three faces.

PYRAMIDS.

652. If a polyedral is cut by a plane which cuts its several edges, the section is a polygon, and a portion of space is cut off, which is called a pyramid.

A PYRAMID is a polyedron having for one face any

polygon, and for its other faces, triangles whose vertices meet at one point.

The polygon is the base of the pyramid, the triangles are its sides, and their intersections are the lateral edges of the pyramid. The vertex of the polyedral is the vertex of the pyramid, and the perpendicular distance from that point to the plane of the base is its altitude.

Pyramids are called triangular, quadrangular, pentagonal, etc., according to the polygon which forms the base. The tetraedron is a triangular pyramid.

653. Problem.—Every pyramid can be divided into the same number of tetraedrons as its base can be into triangles. Let a diagonal plane pass through the vertex of the pyramid and each diagonal of the base, and the solution is evident.

EQUAL PYRAMIDS.

654. Theorem.—-Two pyramids are equal when the base and two adjacent sides of the one are respectively equal to the corresponding parts of the other, and they are similarly arranged.

For the triedrals formed by the given faces in the two must be equal, and may therefore coincide; and the given faces will also coincide, being equal. But now the vertices and bases of the two pyramids coincide. These include the extremities of every edge. Therefore, the edges coincide; also the faces, and the figures throughout.

SIMILAR PYRAMIDS.

655. Theorem. Two similar pyramids are composed of tetraedrons respectively similar, and similarly arranged; and, conversely, two pyramids are similar when composed of similar tetraedrons, similarly arranged.