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6.

One plane is perpendicular to another plane when any straight line drawn in one of the planes perpendicular to the common section is also perpendicular to the other plane.

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Thus in the adjoining figure, the plane EB is perpendicular to the plane CD, if any straight line PQ, drawn in the plane EB at right angles to the common section AB, is also at right angles to the plane CD.

7. The inclination of a plane to a plane is the acute angle contained by two straight lines drawn from any point in the common section at right angles to it, one in one plane and one in the other.

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NOTE. This definition assumes that the angle PQR is of constant magnitude whatever point Q is taken in AB: the truth of which assumption is proved in Proposition 10.

The angle formed by the intersection of two planes is called a dihedral angle.

It may be proved that two planes are perpendicular to one another when the dihedral angle formed by them is a right angle.

8. Parallel planes are such as do not meet when produced.

9. A straight line is parallel to a plane if it does not meet the plane when produced.

10. The angle between two straight lines which do not meet is the angle contained by two intersecting straight lines respectively parallel to the two non-intersecting lines.

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11. A solid angle is that which is made by three or more plane angles which have a common vertex, but are not in the same plane.

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12. A solid figure is any portion of space bounded by one or more surfaces, plane or curved.

These surfaces are called the faces of the solid, and the intersections of adjacent faces are called edges.

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POLYHEDRA.

13. A polyhedron is a solid figure bounded by plane faces.

Obs. A plane rectilineal figure must at least have three sides; or four, if two of the sides are parallel. A polyhedron must at least have four faces; or, if two faces are parallel, it must at least have five faces.

14. A prism is a solid figure bounded by plane faces, of which two that are opposite are similar and equal polygons in parallel planes, and the other faces are parallelograms.

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The polygons are called the ends of the prism. A prism is said to be right if the edges formed by each pair of adjacent parallelograms are perpendicular to the two ends; if otherwise the prism is oblique.

15. A parallelepiped is a solid figure bounded by three pairs of parallel plane faces.

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A parallelepiped may be rectangular as in fig. 1, or oblique as in

fig. 2.

16. A pyramid is a solid figure bounded by plane faces, of which one is a polygon, and the rest are triangles having as bases the sides of the polygon, and as a common vertex some point not in the plane of the polygon.

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The polygon is called the base of the pyramid.

A pyramid having for its base a regular polygon is said to be right when the vertex lies in the straight line drawn perpendicular to the base from its central point (the centre of its inscribed or circumscribed circle).

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19.

Similar polyhedra are such as have all their solid angles equal, each to each, and are bounded by the same number of similar faces.

20.

A Polyhedron is regular when its faces are similar and equal regular polygons.

21. It will be proved (see page 425) that there can only be five regular polyhedra.

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