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PROPOSITION XII. THEOREM 487. If two straight lines are cut by parallel planes, their corresponding segments are proportional.

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Hyp. Parallel planes MN, PQ, and RS are intersected by two straight lines in A, B, C and D, E, F, respectively.

To prove AB: BC=DE: EF.

Proof. Draw AF, and pass a plane through AC and AF, intersecting the planes PQ and RS in BG and CF, respectively. Then CF is II to BG.

(478) By passing a plane through AF and DF, it follows, similarly,

GE is II to AD.
Hence

AB_ AG
BC GF

(Why ?) DE_AG and EFGF

(Why?) * BCEF

(Ax. 1.) Q.E.D. 488. CoR. If through any point straight lines are drawn intersected by two parallel planes, their corresponding segments are proportional.

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PROPOSITION XIII. THEOREM 489. Planes perpendicular to the same straight line are parallel to each other.

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Hyp. Planes MN and PQ are perpendicular to line AB.
To prove

MN is II to PQ.
Hint. — Apply the indirect method and (98).

PROPOSITION XIV. THEOREM 490. A straight line perpendicular to one of two parallel planes is perpendicular to the other also.

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Hyp. Plane MN is II to plane PQ, and AB is I to plane MN.

To prove

AB is I to PQ. . Proof. Through AB pass any plane intersecting MN in AC and PQ in BD, respectively. AC is II to BD,

(478) and AC is I to AB.

(461) Hence

BD is I to AB. Therefore AB is perpendicular to any line in PQ passing through B. Whence AB is I to plane PQ.

Q.E.D.

(88)

491. Cor. Through a given point, one plane and only one may be passed parallel to a given plane.

Ex. 1046. If two planes are parallel to a third plane, they are parallel to each other.

DIEDRAL ANGLES 492. DEF. A diedral angle is the opening between two planes intersecting in a straight line.

The edge is the line of intersection, and the faces are the intersecting planes.

Thus, in the diedral angle formed by the planes AC and BE, BC is the edge and AC and BE are the faces.

493. A diedral angle may be designated by two letters on its edge; or if several diedral angles have a common edge, by four letters, one on each face, and two on the edge, the letters on the edge being placed between the others.

Thus, the diedral angle in the annexed diagram may be designated by BC or ABCE.

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494. DEF. The plane angle of a diedral angle is the angle formed by perpendiculars to the edge at some point, one in each face.

Thus, CDE is a plane angle of diedral angle AB if CD is perpendicular to AB and AT ED is perpendicular to AB.

Two plane angles of the same diedra) angle, as CDE and FGH, are equal, since their sides are respectively parallel.

Or, the plane angle of a died val angle is op the same, at whatever point of the edge it is bk drawn.

The size of a diedral angle does not depend upon the extent of its faces, but upon the difference of their positions.

495. Two diedral angles are equal when they can be made to coincide.

By superposing two equal diedral angles, their plane angles can be made to coincide.

496. A diedral angle is acute, right, obtuse, or straight, according as its plane angle is acute, right, obtuse, or straight.

It is evident that the two faces of a straight diedral angle ie in the same plane.

Diedral angles are complementary, supplementary, adjacent, etc., according as their plane angles are complementary, supplementary, adjacent, etc.

If a plane meets another plane so as to make the two adjacent diedrals equal, each diedral is evidently a right one.

497. Planes forming right diedral angles are said to be perpendicular to each other. Thus, plane CD is perpendicular to plane AB if they form right diedral angles.

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498. DEF. The projection of a point on a plane is the foot of the perpendicular drawn from that point upon the plane.

The projection of a line upon a plane is the locus of the projection of all the points of the line.

PROPOSITION XV. THEOREM 499. Two diedral angles are equal if their plane angles are equal.

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Hyp. The diedral angles AB and A'B' have equal plane angles CBD and C'B'D' respectively.

To prove diedral angle AB= diedral angle A'B'.
Proof. Since AB is I to BD and I to BC,

AB is I to plane BDC.
And, similarly, A'B' is I to plane B'D'C'.
Apply plane Z DBC to Z D'B'C'.
Then:

B coincides with B',
and plane BDC coincides with plane B'D'C".
Hence
BA coincides with B'A'.

(471) Since

BA coincides with B'A',

DB coincides with D'B',
plane AD coincides with plane A'D'.

and

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