For if through two points of the intersection we pass a straight line, all the points of that line must be common to the two planes (1); and no point without this line can be common to the two planes (8). Hence, the straight line must be the line of intersection of the planes (2).

PROPOSITION II.

13. THEOREM. Only one perpendicular can be drawn from a given point to a plane.

Let QP be a perpendicular to the plane MN, drawn from P, a point in the plane; and let TP be any other line drawn from P; then TP cannot be perpendicular to MN.

For, pass through QP and PT a plane, intersecting MN in the line HK; then, since QP is perpendicular to HK (4), TP cannot be perpendicular to HK (350); neither can TP be perpendicular to MN (4).

M

B

N

Again, let AB be a perpendicular to the plane MN, drawn from A, a point without the plane; and let AC be any other line drawn from A to the plane; then AC cannot be perpendicular to MN.

Draw the line BC. Now, since AB is perpendicular to the plane MN, it is also perpendicular to the line BC (4); and therefore AC cannot be perpendicular to BC (350); hence it cannot be perpendicular to MN (4). Q. E. D.

PROPOSITION III.

14. THEOREM. All perpendiculars to a straight line at a given point lie in a plane which is perpendicular to the line at the given point.

M

B

A

N

Let any line AE be drawn perpendicular to the line AB at a given point A; and let MN be a plane passing through A, and perpendicular to AB; then will AE lie in MN.

For, if AE does not lie in MN, pass through AE and AB a plane BAE; and let AE' be the intersection of this plane with MN. Now, AB is perpendicular to AE' (4), and AE was drawn perpendicular to AB; hence, at A there are two perpendiculars to the line AB, lying in the same plane BAE. But this is impossible. Hence the assumption that AE does not lie in MN is false; or, AE lies in MN. Q. E. D. 15. COROLLARY. Through a given point in a straight line only one plane may be passed perpendicular to the line.

PROPOSITION IV.

16. THEOREM. If a line is perpendicular to each of two lines at their point of intersection, it is perpendicular to the plane of the two lines.

A

M

B

N

Let the line AB be perpendicular to the lines BC and BD at B, their intersection; and let MN be the plane of BC and BD; then will AB be perpendicular to MN.

By our hypothesis BC and BD are perpendiculars to AB, at a common point B; therefore they lie in a plane which is perpendicular to AB at B (14). But, there is but one plane in which both BC and BD can lie; (11), which is by hypothesis MN. Hence MN is perpendicular to AB; or, AB is perpendicular to MN. Q. E. D.

i.e., the plane of the lines

PROPOSITION V.

17. THEOREM. Parallel lines included between two parallel planes are equal.

Let the parallel lines AA', BB', and CC' be included between the parallel planes MN and PQ; then will AA', BB', and CC' be equal.

Draw the lines AC and A'C'. Now these lines are parallel. For, they lie in the same plane, i.e., the plane of the parallel lines AA′ and CC' (11); and they cannot meet, since they lie in the parallel planes MN and PQ (5); hence they are parallel.

AC' is therefore a parallelogram (371); and AA' is equal to CC' (360). In like manner we may prove that AA' equals BB', and that BB' equals CC'. Q. E. D.

18. COROLLARY 1. The intersections of a plane with two parallel planes are parallel lines.

We have shown that AC and A'C' are parallel; i.e., the intersections of the plane AC' with the parallel planes MN and PQ are parallel lines.

19. COROLLARY 2. Angles lying in different planes having their sides parallel and in the same direction each to each, are equal.

We have shown that AC' is a parallelogram; therefore AC is equal and parallel to A'C'. In like manner it may be shown that AB is equal and parallel to A'B', and that BC is equal and parallel to B'C'. It follows then that the triangle ABC is equal to the triangle A'B'C' (357), in which triangles, any angle ABC is equal to the corresponding angle A'B'C'. And these angles have their sides parallel, each to each, and lying in different planes.

20. COROLLARY 3. Lines parallel to the same line are parallel to each other.

For, from the parallelograms AC' and AB', we derive that, the lines CC' and BB' are each parallel to the same line AA'; and they are parallel to each other, since BC' is also a parallelogram.

PROPOSITION VI.

21. THEOREM. If a straight line is perpendicular to one of two parallel planes, it is perpendicular to the other also.

Let MN and PQ be parallel planes, and let AD be perpendicular to PQ; then will AD also be perpendicular to MN.

Through AD pass two planes whose intersections with MN and PQ are the lines DF, AC, and DE, AB. Now DF and AC are parallel (18); and since ADF is a right angle (4), DAC is also a right angle (368). may prove that DAB is a right angle. perpendicular to both AC and AB at section, AD is also perpendicular to MN (16).

In like manner we Hence, since DA is their point of interQ. E. D.

22. COROLLARY. If two planes are perpendicular to the same straight line, they are parallel to each other.

Let the two planes MN and PQ be perpendicular to the straight line AD at the points A and D; then will these planes be parallel to each other.

For, pass through A a plane which is parallel to PQ, and it will be perpendicular to AD (21). But only one plane can be passed through A which is perpendicular to AD (15). Therefore MN must coincide with this plane which has been passed through A, parallel to PQ; i.e., MN is parallel to PQ.

PROPOSITION VII.

THEOREM. If one of two parallel lines is perpendicu lar to a plane, the other is also perpendicular to the plane.

Let the line BA be perpendicular to the plane xy; and let the line DC be parallel to BA; then will DC be perpendicular to xy.

In xy, draw through C, any line CF, and parallel to CF draw AE; then the angles BAE and DCF will be equal (19). But BAE is a right angle (4); therefore DCF is a right angle. Then, since DC is perpendicular to CF any line in the plane xy, DC must also be perpendicular to xy (4). Q. E. D.

24. COROLLARY 1. If two lines are perpendicular to the same plane they are parallel to each other.

Let the lines BA and DC be perpendicular to the plane. xy at the points A and C. Now through A, draw a parallel