Proposition 10. Parallel Planes Intersected

55. Theorem. If two parallel planes are cut by a third plane, the lines of intersection are parallel.

B

m

n

Given the

planes m and n, intersected by a third plane p

in AB and CD respectively.

Prove that

Proof.

AB is to CD.

AB and CD are in the same plane p.

Since AB is always in m and CD is always in n, m and n must meet if AB and CD meet.

But

and hence

m is Il to n,

m and n cannot meet.

.. AB is to CD.

Given

$ 3,9

Given

$36

$ 9, 1

56. Corollary. A line perpendicular to one of two parallel planes is perpendicular to the other also.

Let PQ be to m, and let p and q be two planes containing PQ. Now prove (§ 55) that QB is to PA and that QD is I to PC. Then prove that PQ is 1 to QB and QD, and hence that PQ is to n.

n p

Q

B

57. Distance between Parallel Planes. The length of a perpendicular line segment between two parallel planes is called the distance between the planes.

It has been shown (§ 56) that if this segment is perpendicular to one of two parallel planes it is perpendicular to the other. It will now be shown (§ 58) that the length is the same whatever perpendicular between the planes is taken.

58. Corollary. Two parallel planes are everywhere equidistant from each other.

In the figure of § 56, if AB is constructed II to PQ, then ABQP is a (§ 10, 2). Since it is given that PQ is to m, then AB is also to m (§ 48). Both AB and PQ are then I to n (§ 56) and represent distances measured on any two s. But AB=PQ (§ 10, 3), and hence m and n are everywhere equidistant from each other.

59. Parallel in the Same Sense. If two parallel rays lie on the same side of the line segment which joins their end points, they are said to be parallel in the same sense.

Exercises. Parallel Planes

1. Parallel lines included between parallel planes are equal.

2. The locus of points equidistant from two parallel planes is a plane which is perpendicular to a line perpendicular to the planes and which bisects the segment cut off by them.

3. The locus of points equidistant from two parallel lines is a plane which is perpendicular to a line perpendicular to the given lines and which bisects the segment cut off by them.

4. The locus of points at a given distance from a plane is a pair of planes, each at the given distance from the given plane and parallel to it.

Proposition 11. Arms of Angles Parallel

60. Theorem. If two angles not in the same plane have their arms respectively parallel in the same sense, the angles are equal and their planes are parallel.

Given the A and A' in the planes m and n respectively, with their arms respectively || in the same sense.

Prove that LA=ZA' and that m is || to n.

Proof. Take AB=A'B', and AC=A'C', and draw BC, B'C', AA', BB', CC'.

By proving that A ́B ́BA and A'C'CA are B'BCC' is a □, and hence that BC= B'C' (§ 10, 3).

Then

(§ 10, 6), show that

$7,5

$ 7,2

AABC is congruent to ▲ A'B'C'.
..ZA=ZA'.

$ 32

Now if m is not I to n, they will meet in a line l. Since AB and AC are || to n (§ 50), neither can meet l. But since both AB and AC cannot be || to 7 (§ 49), m cannot meet n, and hence m is I to n.

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

In the figure of § 60, if AB and AC are both II to n, show that AB cannot meet a line A'B', which is the intersection of n and the plane of AA' and AB. Similarly, AC is || to A'C'. Hence m is || to n (§ 60).

Proposition 12. Transversals in Space

62. Theorem. If two lines are cut by three parallel planes, their corresponding segments are proportional.

Given AB and CD, cut by the || planes p, q, r, in the points A, E, B, and C, F, D respectively.

Proof. Let q intersect the plane of A, B, D in EG and the plane of A, D, C in GF.

It should be observed that this proposition is a generalization of § 22,7. It may also be stated as follows: If two lines are cut by any number of parallel planes, their corresponding segments are proportional.

Consider also the case where AB and CD, although not lying in a plane, appear to cross between the planes p and r.

S

Exercises. Review

1. In a given plane find the locus of points equidistant from two given points not in the plane.

2. Find the locus of points equidistant from three given points not in a straight line.

3. Find the locus of points equidistant from two given parallel planes and also equidistant from two given points. 4. What is the locus of points at a given distance from each of two planes?

5. The line AB cuts three parallel planes in the points A, E, B; and the line CD cuts these planes in the points C, F, D. If AE 3 in., EB=4 in., and CD = 6 in., what are the lengths of CF and FD?

=

6. In Ex. 5, if AB=16 in., CF10 in., and CD = 18 in., what are the lengths of AE and EB?

7. It is proved in plane geometry that if three or more parallels intercept equal segments on one transversal, they intercept equal segments on every transversal. State and prove a corresponding proposition in solid geometry.

8. It is proved in plane geometry that the line which joins the midpoints of two sides of a triangle is parallel to the third side. State and prove a proposition in solid geometry which shall refer to a plane passing through the midpoints of two sides of a triangle.

9. A cylindric water tank which is 16 ft. deep and 12 ft. in diameter is filled with water to a depth of 9 ft. A pole standing obliquely in the tank just reaches from a point on the circumference of the base to a point exactly opposite on the upper rim. Find the length of that part of the pole which is under water.

10. Consider Ex. 9 when the water level rises 3 ft.