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DIHEDRAL ANGLES.

536. DEF. The opening between two intersecting planes is called a dihedral angle.

537. DEF. The line of intersection AB of the planes is the edge, the planes MA and NB are the faces, of the dihedral angle.

538. A dihedral angle is designated by its edge, or by its two faces and its edge. Thus, the dihedral angle in the margin may be designated by AB, or by M-AB-N.

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539. In order to have a

clear notion of the magni

tude of the dihedral angle

M-AB-N, suppose a plane at first in coincidence with the plane MA to turn about the edge AB, as indicated by the arrow, until it coincides with the plane NB. The magnitude of the dihedral angle M-AB-N is proportional to the amount of rotation of this plane.

540. DEF. Two dihedral angles M-AB-N and P-AB-N are adjacent if they have a

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jacent dihedral angles equal, each of these angles is called a right dihedral angle.

542. DEF. A plane is perpendicular to another plane if it forms with this second plane a right dihedral angle.

543. DEF. Two vertical dihedral angles are dihedral angles that have the same edge and the faces of the one are the prolongations of the faces of the other.

544. DEF. Dihedral angles are acute, obtuse, complementary, supplementary, under the same conditions as plane angles.

545. DEF. The plane angle of a dihedral angle is the plane angle formed by two straight lines, one in each plane, perpendicular to the edge at the same point.

546. COR. The plane angle of a dihedral angle has the same magnitude from whatever point in the edge

the perpendiculars are drawn.

For any two such angles, as CAD, GIH, have their sides respectively parallel (§ 104), and hence B are equal.

§ 534

C

H

G

F

E

547. The demonstrations of many properties of dihedral angles are identically the same as the demonstrations of analogous theorems of plane angles.

The following are examples:

1. If a plane meets another plane, it forms with it two adjacent dihedral angles whose sum is equal to two right dihedral angles.

2. If the sum of two adjacent dihedral angles is equal to two right dihedral angles, their exterior faces are in the same plane.

3. If two planes intersect each other, their vertical dihedral angles are equal.

4. If a plane intersects two parallel planes, the alternateinterior dihedral angles are equal; the exterior-interior dihedral angles are equal; the two interior dihedral angles on the same side of the transverse plane are supplementary.

5. When two planes are cut by a third plane, if the alternateinterior dihedral angles are equal, or the exterior-interior dihedral angles are equal, and the edges of the dihedral angles thus formed are parallel, the two planes are parallel.

6. Two dihedral angles whose faces are parallel each to each are either equal or supplementary.

PROPOSITION XVI. THEOREM.

548. Two dihedral angles are equal if their plane angles are equal.

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Let the two plane angles ABD and A ́B ́D ́ of the two dihedral angles D-CB-E and D'-C ́B ́-E' be equal.

To prove the dihedral angles D-CB-E and D'-C'B'-E' equal.

Proof. Apply D'-C'B'-E' to D-CB-E, making the plane angle A'B'D' coincide with its equal ABD.

The line B'C' being to the plane A'B'D' will likewise be I to the plane ABD at B, and fall on BC, since at B only one can be erected to this plane.

§ 511

The two planes A'B'C' and ABC, having in common two

intersecting lines AB and BC, coincide.

§ 497

In like manner the planes D'B'C' and DBC coincide. Therefore, the two dihedral angles coincide and are equal.

Q. E. D

PROPOSITION XVII. THEOREM.

549. Two dihedral angles have the same ratio as their plane angles.

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Let A-BC-D and A-B'C'-D' be two dihedral angles, and let their plane angles be ABD and A'B'D', respectively.

To prove that A'-B'C'-D': A-BC-D = LA'B'D': Z ABD. CASE 1. When the plane angles are commensurable. Proof. Suppose the ABD and A'B'D' (Figs. 1 and 2) have a common measure, which is contained m times in and n times in A'B'D'.

Then

LA'B'D': L ABD = n: m.

ABD

Apply this measure to ABD and A'B'D', and through the lines of division and the edges BC and B'C' pass planes. These planes divide A-BC-D into m parts, and A'-B'C'-D' into n parts, equal each to each.

Therefore, A-B'C'-D': A-BC-D=n: m.

§ 548

Therefore, A'-B'C'-D': A-BC-DL A'B'D': Z ABD. Ax. 1

CASE 2. When the plane angles are incommensurable.

Proof. Divide the ABD into any number of equal parts, and apply one of these parts to the A'B'D' (Figs. 1 and 3) as a unit of measure.

Since ABD and ▲ A'B'D' are incommensurable, a certain number of these parts will form the A'B'E, leaving a remainder EB'D', less than one of the parts.

Pass a plane through B'E and B'C'.

Since the plane angles of the dihedral angles A-BC-D and A-B'C'-E are commensurable,

A'-B'C'-E: A-BC-D = LA'B'E:L ABD. By increasing the number of equal parts into which

Case 1
ABD

is divided, we can diminish at pleasure the magnitude of each

part, and therefore make

value, however small, since

EB'D' less than any assigned

EB'D' is always less than one

of the equal parts into which ▲ ABD is divided.

But we cannot make pothesis ABD and

EB'D' equal to zero, since by hyA'B'D' are incommensurable.

§ 269

Therefore, EB'D' approaches zero as a limit, if the number of parts into which ABD is divided is indefinitely increased; and the corresponding dihedral angle E-B'C'-D' approaches zero as a limit.

§ 275

Therefore, ▲ A'B'E approaches ▲ A'B'D' as a limit, § 271 A'-B'C'-E approaches A'-B'C'-D' as a limit.

and

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as EB'D' varies in value and approaches zero as a limit. Therefore, the limits of these variables are equal.

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§ 284

Q. E. D.

550. COR. The plane angle of a dihedral angle may be taken as the measure of the dihedral angle.

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