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PROPOSITION XIX. THEOREM.

475. If two intersecting planes be each perpendicular to a third plane, their intersection is also perpendicular to that plane.

P

B

D

Let the planes BD and BC intersecting in the line AB be perpendicular to the plane PQ.

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A perpendicular erected at B, a point common to the three planes, will lie in the two planes BC and BD,

$ 473

(if two planes be to each other, a straight line drawn through any point to one of the planes will lie in the other plane).

of intersection

And, since this

lies in both the planes, BC and BD, it

must coincide with their intersection.

.. AB is to the plane PQ.

Q. E. D.

476. COROLLARY. If a plane be perpendicular to each of two intersecting planes, it is perpendicular to the intersection of those planes.

PROPOSITION XX. THEOREM.

477. Every point in the plane which bisects a dihedral angle is equally distant from the faces of that angle.

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Let plane A M bisect the dihedral angle formed by the planes AD and AC; and let PE and PF be perpendiculars drawn from any point P in the plane A M to the planes AC and A D.

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Through PE and PF pass a plane intersecting the planes A C and AD in O E and O F.

A D,

Join PO.

Now the plane PEF is to each of the planes AC and

§ 471 (if a straight line be to a plane, any plane embracing the line is to that

plane);

.. the plane PEF is to their intersection A O. § 476 (If a plane be to each of two intersecting planes, it is to the intersection of these planes).

..L POEL POF,

(being measures respectively of the equal dihedral & M-OA-C and M-0A-D).

.. rt. A POE = rt. ▲ PO F,

.. PE=PF,

(being homologous sides of equal ▲).

§ 110

Q. E. D.

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478. The acute angle which a straight line makes with its own projection on a plane is the least angle which it makes with any line of that plane.

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A

N

Let BA meet the plane M N at B, and let BA' be its projection upon the plane MN, and BC any other line drawn through B in the plane.

We are to prove ▲ABA' <▲ A B C.

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(a is the shortest distance from a point to a plane).

.. ZABA' < LA BC,

Iden.

Cons.

§ 448

$ 116

(if two sides of a ▲ be equal respectively to two sides of another, but the third side of the first ▲ be greater than the third side of the second, then the Zopposite the third side of the first ▲ is greater than the third side of the second).

opposite the

Q. E. D.

EXERCISE. The angle included by two perpendiculars drawn from any point within a dihedral angle to its faces, is the supplement of the dihedral angle.

PROPOSITION XXII. THEOREM.

479. If two straight lines be not in the same plane, one and only one common perpendicular to the lines can be drawn.

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Let A B and C D be two given straight lines not in the same plane.

We are to prove one and only one common perpendicular to the two lines can be drawn.

not ll,

Since AB and CD are not in the same plane they are

(two Ils lie in the same plane).

§ 474

Through the line A B pass the plane M N l to CD. Since CD is to the plane MN, all its points are equally distant from the plane MN;

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hence CD', the projection of the line CD on the plane M N, will be II to CD,

$76

and will intersect the line A B at some point as C". Now since CC' is the line which projects the point C upon the plane MN, it is to the plane MN;

to C'D' and A B,

§ 435

§ 430

to every line drawn through its foot in the plane).

hence CC is

(if a line be to a plane, it is

Also, CC' is to CD,

§ 67

.. C C' is the common

to the lines CD and A B.

be

Moreover, line C C is the only common 1.

For, if another line E B, drawn between A B and C D, could to AB and C D, it would also be to a line BG drawn

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.. C C is the only common to the lines CD and A B.

Q. E. D.

ON POLYHEDRAL ANGLES.

480. DEF. A Polyhedral angle is the extent of opening of

three or more planes meeting in a common point.

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The plane angles ASB, BSC, etc., formed by the edges are its face angles.

481. DEF. Polyhedral angles are classified as trihedral, quadrahedral, etc., according to the number of the faces.

482. DEF. Trihedral angles are rectangular, bi-rectangular, or tri-rectangular, according as they have one, two, or three right dihedral angles.

483. DEF. Trihedral angles are scalene, isosceles, or equilateral, according as the face angles are all unequal, two equal, or three equal.

484. DEF. A polyhedral angle is convex, if the polygon formed by the intersections of a plane with all its faces be a convex polygon.

485. DEF. Two polyhedral angles are equal when they can be applied to each other so as to coincide in all their parts.

Since two equal polyhedral angles coincide however far their edges and faces be produced, the magnitude of a polyhedral angle does not depend upon the extent of its faces. But, in order to represent the angle in a diagram, it is usual to pass a plane, as

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