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530. COR. 2. Two parallel planes are everywhere equally distant.

For is dropped from any points in MN to PQ measure the distances of these points from PQ. But these Is are parallel (§ 519), and hence equal (§ 529). Therefore, all points in MN are equidistant from PQ.

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531. A straight line perpendicular to one of two parallel planes is perpendicular to the other also.

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Let AB be perpendicular to MN and PQ parallel to MN.

To prove that AB is perpendicular to PQ.

Proof. Pass through the line AB any two planes intersecting MN in the lines AC and AD, and PQ in BE and BF. Then AC and AD are to BE and BF, respectively.

§ 528

But AB is 1 to AC and AD.

§ 501

.. AB is to their parallels BE and BF.
Therefore, AB is 1 to PQ.

§ 107

§ 507

Q. E. D.

532. COR. Through a given point one plane, and only one, can be drawn parallel to a given plane.

For if a line is drawn from A to PQ, a plane passing through to this line is to PQ (§ 527); and since through a point in a line only one plane can be drawn to the line (§ 509), only one plane can be drawn through A || to PQ.

PROPOSITION XIII. THEOREM.

533. If two intersecting straight lines are each parallel to a plane, the plane of these lines is parallel to that plane.

M

D

B

Let AC and AD be each parallel to the plane PQ, and let MN be the plane passed through AC and AD.

To prove that MN is parallel to PQ.

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Pass a plane through AB and AC intersecting PQ in BE, and a plane through AB and AD intersecting PQ in BF.

Then AB is to BE and BF.
Also, BE is to AC, and BF is || to AD.

§ 501

§ 525

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Ex. 606. Find the locus of all lines drawn through a given point, parallel to a given plane.

Ex. 607. Find the locus of points in a given plane which are equidistant from two given points not in the plane.

Ex. 608. Find the locus of a point in space equidistant from three given points not in a straight line.

Ex. 609. Find a point in a plane such that the sum of its distances from two given points on the same side of the plane shall be a minimum.

PROPOSITION XIV. THEOREM.

534. If two angles not in the same plane have their sides respectively parallel and lying on the same side of the straight line joining their vertices, they are equal, and their planes are parallel.

M

Ci

N

Q

Let the angles A and A' be respectively in the planes MN and PQ, and have AD parallel to A ́D' and AC parallel to A'C' and lying on the same side of AA'.

To prove that ▲ A

=

ZA', and that MN is to PQ.

Proof. Take AD and A'D' equal, also AC and A'C' equal.

Draw DD', CC', CD, C'D'.

Since AD is equal and to A'D', the figure ADD'A' is a parallelogram, and AA' is equal and || to DD'.

In like manner AA' is equal and I to CC'.

§ 183

Also, since CC' and DD' are each || to AA', and equal to AA', they are and equal.

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Now PQ is to each of the lines AC and AD.

Therefore, PQ is | to MN, the plane of these lines.

§ 183

§ 150

§ 128

$ 522

§ 533

Q. E. D.

PROPOSITION XV. THEOREM.

535. If two straight lines are intersected by three parallel planes, their corresponding segments are propor

tional.

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Let AB and CD be intersected by the parallel planes MN, PQ, RS, in the points A, E, B, and C, F, D.

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Ex. 610. The line AB meets three parallel planes in the points A, E, and the line CD meets the same planes in the points C, F, D. If AE 6 inches, BE 8 inches, CD 12 inches, compute CF and FD.

B;

=

=

Ex. 611. To draw a perpendicular to a given plane from a given point without the plane.

Ex. 612. To erect a perpendicular to a given plane at a given point in the plane.

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

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dihedral angle in the margin may be designated by AB, or by M-AB-N.

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 common edge AB, and a common face NB, between them.

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A

N

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.

и

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