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EXERCISES.

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

2. If each of the projections of a line A B upon two intersecting planes be a straight line, the line A B is a straight line.

3. The height of a room is eight feet, how can a point in the floor directly under a certain point in the ceiling be determined with a ten-foot pole?

4. If a line be drawn at an inclination of 45° to a plane, what is the greatest angle which any line of the plane, drawn through the point in which the inclined line pierces the plane, makes with the line.

5. Through a given point pass a plane parallel to a given plane.

6. Find the locus of points in space which are equally distant from two given points.

7. Show that the three planes embracing the edges of a trihedral angle and the bisectors of the opposite face angles respectively intersect in the same straight line.

8. Find the locus of the points which are equally distant from the three edges of a trihedral angle.

9. Cut a given quadrahedral angle by a plane so that the section shall be a parallelogram.

10. Determine a point in a given plane such that the sum of its distances from two given points on the same side of the plane shall be a minimum.

11. Determine a point in a given plane such that the difference of its distances from two given points on opposite sides of a plane shall be a maximum.

PROPOSITION XXVII. THEOREM.

492. Two trihedral angles are equal or symmetrical when the three face angles of the one are respectively equal to the three face angles of the other.

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In the trihedral S and S', let ▲ AS B

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LASC

=

ZA'S'C', and BSC

= LB'S'C'.

We are to prove that the homologous dihedral angles are equal, and hence the trihedral angles S and S' are either equal or symmetrical.

On the edges of these angles take the six equal distances SA, SB, SC, S' A', S' B', S' C'.

Draw A B, BC, A C, A'B', B'C', A'C'.

The homologous isosceles ASA B, SA'B', SA C, S' A' C',

SBC, S'B'C' are equal, respectively.

.. AB, A C, B C equal respectively A' B', A' C', B' C',

(being homologous sides of equal ▲).

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$106

§ 108

At any point D in SA draw DE and DFL to SA in the

faces ASB and A SC respectively.

These lines meet A B and A C respectively,

(since the SA B and S AC are acute, each being one of the equal of an

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Draw D'E' and D' F in the faces A' S'B' and A' S′ Ç' respectively to S' A', and join E' F'.

In the rt. A A DE and A' D' E'

AD=A' D',

=

LDAE L D'A' E',

(being homologous of the equal & SAB and S' A' B').

.. rt. A ADE = rt. ▲ A' D' E',

Cons.

§ 111

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(being homologous sides of equal ▲).

In like manner we may prove A F

=

A'F' and D F = D' F'.

Hence in the AA EF and A' E' F' we have

A E and A F =

and

respectively A' E' and A' F',

LEAF LEAF,

=

(being homologous of the equal & ABC and A'B'C').

.. AAEF=▲ A' E' F',

.. EF = EF

(being homologous sides of the equal & A EF and A' E' F').

Hence, in the A EDF and E'D' F' we have

=

§ 106

ED, DF, and E F respectively E' D', D' F', and E' F'.

.. AEDF ▲ E' D' F',

=

LEDFL E' D' F',

(being homologous of equal ▲).

..the dihedral ▲ B-A S-C = dihedral

B'-A' S'-C',

§ 108

(since & ED F and E' D' F", the measures of these dihedral 4, are equal).

In like manner it may be proved that the dihedral A-B S-C and A-C SB are equal respectively to the dihedral A'-B' S'-C and A'-C' S'-B'.

Q. E. D.

This demonstration applies to either of the two figures denoted by S'-A'B'C', which are symmetrical with respect to each other. If the first of these figures be given, S and S' are equal, for they can be applied to each other so as to coincide in all their parts. If the second be given, S and S are symmetrical. § 486

BOOK VII.

POLYHEDRONS, CYLINDERS, AND CONES.

GENERAL DEFINITIONS.

493. DEF. A Polyhedron is a solid bounded by four or more polygons.

A polyhedron bounded by four polygons is called a tetrahedron; by six, a hexahedron; by eight, an octahedron; by twelve, a dodecahedron; by twenty, an icosahedron.

494. DEF. The Faces of a polyhedron are the bounding polygons.

495. DEF. tions of its faces.

The Edges of a polyhedron are the intersec

496. DEF. The Vertices of a polyhedron are the intersections of its edges.

497. DEF. A Diagonal of a polyhedron is a straight line joining any two vertices not in the same face.

498. DEF. A Section of a polyhedron is a polygon formed by the intersection of a plane with three or more faces.

499. DEF. A Convex polyhedron is a polyhedron every section of which is a convex polygon.

500. DEF. The Volume of a polyhedron is the numerical measure of its magnitude referred to some other polyhedron as a unit of measure.

501. DEF. The polyhedron adopted as the unit of measure is called the Unit of Volume.

502. DEF. Similar polyhedrons are polyhedrons which have the same form.

503. DEF. Equivalent polyhedrons are polyhedrons which have the same volume.

504. DEF. Equal polyhedrons are polyhedrons which have the same form and volume.

ON PRISMS.

505. DEF. A Prism is a polyhedron two of whose faces are equal and parallel polygons, and the other faces are parallelo

grams.

506. DEF. The Bases of a prism are the equal and parallel polygons.

507. DEF. The Lateral faces of a prism are all the faces except the bases.

508. DEF. vex Surface of a its lateral faces.

The Lateral or Con-
prism is the sum of

509. DEF. The Lateral edges of a prism are the intersections of its lateral faces; the Basal edges of a prism are the intersections of the bases with the lateral faces.

OBLIQUE PRISM.

510. DEF. Prisms are triangular, quadrangular, pentangular, etc., according as their bases are triangles, quadrangles, pentagons, etc.

511. DEF. A Right prism is a prism whose lateral edges are perpendicular to its bases.

512. DEF. An Oblique prism is a prism whose lateral edges are oblique to its bases.

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

514. DEF. The Altitude of a prism is the perpendicular distance between the planes of its bases. The altitude of a right prism is equal to any one of its lateral edges.

515. DEF. 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.

RIGHT PRISM,

516. DEF. A Right section of a prism is a section perpendicular to its lateral edges.

517. DEF. A Parallelopiped is a prism whose bases are parallelograms.

518. DEF. A Right parallelopiped is a parallelopiped whose lateral edges are perpendicular to its bases; hence its lateral faces are rectangles.

519. DEF. An Oblique parallelopiped is a parallelopiped whose lateral edges are oblique to its bases.

520. DEF. A Rectangular parallelopiped is a right parallelopiped whose bases are rectangles.

521. DEF. A Cube is a rectangular parallelopiped all of whose faces are squares.

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