PROPOSITION V. THEOREM.

345. Two straight lines perpendicular to the same plane are parallel.

Let AB and CD be two straight lines perpendicular to the plane MN.

lel.

To prove that AB and CD are paral- M

In the plane MN draw BD and AD

and erect DE perpendicular to BD.

Since DC is perpendicular to the

A

C

plane MN it is (by 326) perpendicular to DE.

Again, since BD is perpendicular to ED, by construction, AD is perpendicular (by 344) to ED.

Therefore ED is perpendicular to AD, BD, and CD; hence

these lines all lie in one plane.

Consequently AB and CD are two lines in one plane perpendicular to the same line BD, therefore (by 60) they are parallel to one another.

Q.E.D.

346. COR. 1. If one of two parallels is perpendicular to a plane, the other is also.

347. COR. 2. Two straight lines that are parallel to a third straight line are parallel to each other.

EXERCISES.

1. Two planes that have three points not in the same straight line in common coincide.

2. At a given point in a plane, erect a perpendicular to the plane.

3. From a point without a plane, let fall a perpendicular to the plane.

4. From a point without a plane draw a number of equal oblique lines to the plane.

348. If a straight line and a plane be perpendicular to the same straight line, they are parallel.

Let the straight line BC and the plane MN be perpendicular to the straight line AB.

To prove that BC is parallel to MN. Pass a plane through BC and A meeting MN in the line AD, then from any point C in the line BC let fall the perpendicular CD, and draw AD. Since CD is perpendicular to MN it will (by 326) be perpendicular to AD.

But BA is perpendicular to AD, therefore (by 345) BA is parallel to CD, and likewise BC and AD being perpendicular to BA, they will be parallel.

Therefore BADC is a parallelogram and CD BA, or the line BC is everywhere equally distant from MN, hence is parallel to MN.

Q.E.D.

349. COR. 1. If two planes be perpendicular to the same straight line, they are parallel.

350. COR. 2.

distant.

Two parallel planes are everywhere equally

351. COR. 3. If two intersecting straight lines are each parallel to a given plane, the plane of these lines is parallel to the given plane.

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

Let MN and PQ be two parallel planes intersected by the plane AD in AB and CD.

To prove that AB and CD are parallel. The lines AB and CD cannot meet since they lie in planes that are parallel, they themselves by hypothesis being in the same plane AD.

Therefore AB and CD are parallel.

EXERCISES.

M

B

A

N

P

D

C

Q.E.D.

1. If a straight line is parallel to a line in a plane, it is parallel to the plane.

2. Parallel lines between parallel planes are equal.

SUGGESTION.

See 108.

PROPOSITION VIII.

THEOREM.

353. If two angles not in the same plane have their sides respectively parallel and lying in the same direction, they are equal and their planes are parallel.

Let the angles C and C' lie in the planes MN and PQ respectively, having their sides AC and A'C' parallel, and also CB and C'B parallel and in the same direction. To prove that ≤ C

and PQ are parallel.

M

C

A

B

1. Take A'C' = AC and C'B' = CB, and

A

draw AA', BB', and CC'.

Since AC and A'C' are equal by con

2

struction and parallel by hypothesis, the figure ACC'A' is a parallelogram; that is, AA' is equal and parallel to CC".

For a similar reason, BB' and CC" are equal and parallel. Since AA' and BB' are both equal and parallel to CC", they are equal and parallel to each other, or ABB'A' is a parallelogram, and hence AB = A'B'.

Therefore the triangles ACB and A'C'B' have their sides equal, and hence their angles are equal, or ≤ C = ≤ C'.

2. Since AA' = BB' = CC', the two planes are equally distant, and hence parallel.

Q.E.D.

354. COR. If two angles have their sides parallel, they are equal or supplemental.

PROPOSITION IX. THEOREM.

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

Let AB and CD be intersected by the parallel planes MN,

PQ, RS, in the points A, E, B, and C, F, D.

In the triangle ABD, EG, being in the plane PG parallel to RS, will be parallel to BD.

Likewise in the triangle DAC, GF is parallel to AC, and hence

356. COR. Any number of straight lines cut by parallel planes are divided into proportional segments.

DIEDRAL ANGLES.

DEFINITIONS.

357. A Diedral Angle is the difference of direction of two planes.

If the planes are produced until they intersect, the line of intersection is called the Edge. The planes are the Faces. Thus in the diedral angle formed by the planes BD and BF, BE is the edge and BD and BF are the faces.

A

B

CG

K

D

E

F

358. A diedral angle may be designated by H two letters on its edge; or, if several diedral angles have a common edge, by four letters, one in each face and two on the edge, the letters on the edge being named between the other two. Thus the diedral angle in the figure may be designated either as BE or ABEC.

359. If a point is taken in the edge of the diedral angle, and two straight lines are drawn through this point, one in