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Let the two parallel lines A B and C D be included between the parallel planes M N and PQ.

We are to prove

If AB and CD be

A B C D.

to the two Il planes they are equal, § 434 (if two planes be ||, all the points of either are equally distant from the other).

If A B and CD be not the points A and C the lines A E and C F

to the two

planes, draw from to the plane M N.

§ 458

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$434

$ 430

(if a straight line be to a plane it is to any line of the plane drawn

and

through its foot);

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=

$462 (if two not in the same plane have their sides || and lying in the same direction they are equal).

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PROPOSITION XII. THEOREM.

465. The intersections of two parallel planes by a third plane are parallel lines.

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Let the plane OS intersect the parallel planes P Q and M N in the lines A C and B D respectively.

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PROPOSITION. XIII. THEOREM.

466. If a straight line be perpendicular to one of two parallel planes it is perpendicular to the other.

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Let MN and PQ be parallel planes and AB be per

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Let two planes embracing AB intersect the planes M N and PQ in AC, BE and A D, B F respectively.

Then

AC is to BE and AD to BF,

§ 465

(the intersections of two || planes by a third plane are || lines).

But EB and FB are (if a straight line be to a plane it is to every straight line of the plane drawn through its foot).

to A B,

$430

.. AC and AD which are respectively to BE and BF

are to A B,

(if a straight line be to one of two || lines, it is to the other).

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$67

§ 449

(if a line be to two straight lines in a plane drawn through its foot it is

to the plane).

Q. E. D.

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467. COROLLARY. If two planes be parallel to a third plane. they are parallel to each other. For, every line perpendicular to this third plane is perpendicular to the other planes; and two

planes perpendicular to a straight line are parallel.

PROPOSITION XIV. THEOREM.

468. If a straight line be parallel to another straight line drawn in a plane, it is parallel to the plane.

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Let AC be parallel to the line B D in the plane M N.

We are to prove AC to the plane M N.

From A and C, any two points in A C, draw A B and C D 1 to A C, and A E and C F 1 to the plane M N.

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(if two & not in the same plane have their sides || and lying in the same

direction, they are equal).

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Now since the points A and C, any two points in the line A C, are equally distant from the plane MN,

all the points in AC are equally distant from the plane MN.

.. A C is to the plane MN.

$432

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Q. E. D.

PROPOSITION XV. THEOREM.

469. If two straight lines be intersected by three parallel planes their corresponding segments are proportional.

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

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(the intersections of two || planes by a third plane are || lines).

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$465

$ 275

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(a line drawn through two sides of a ▲l to the third side divides those

Also,

sides proportionally).

GF is to A C,

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§ 465

$275

Ax. 1.

Q. E. D.

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