same direction, although not in the same plane with the former, the two angles will be equal, and their planes will be parallel. Let the two straight lines AB. BC, in the plane XZ, be parallel to the two DE, EF, in the plane WY; = Then angle ABC= angle DEF. For, make BA = ED, BC= EF; join A, C; D, F; A, D; B, E; Č, F; Then the straight lines AD, BE, which join the equal and parallel straight lines AB, DE, are themselves equal and parallel. Ꮓ W H X For the same reason, CF, BE are equal and parallel. .. AD, CF are equal and parallel (cor., Prop. 10), and .. AC, DF are also equal and parallel (th. 21). Hence the two triangles ABC, DEF, having all their sides equal, each to each, have their angles also equal. ... angle ABC= angle DEF. Again, the plane XZ is parallel to the plane WY. For, if not, let a plane drawn through A, parallel to DEF, meet the straight lines FC, EB in G and H. Then DA= EH-FG (Prop. 13). But DA=EB = FC ... EH = EB, FG=FC, which is absurd. Cor. 1. If two parallel planes XZ, WY are met by two other planes ADEB, CFEB, the angles ABC, DEF, formed by the intersection of the parallel planes, will be equal. For the section AB is parallel to the section DE (Prop. 12). So, also, the section BC is parallel to the section EF. .. angle ABC= angle DEF. Cor. 2. If three straight lines AD, BE, CF, not sit uated in the same plane, be equal and parallel, the triangles ABC, DEG, formed by joining the extremities of these straight lines, will be equal, and their planes will be parallel. PROP. XVII. If two straight lines be cut by parallel planes, they will be cut in the same ratio. Let the straight lines AB, CD be X Join A, C; B, D; A, D; and let Then the intersections EG, BD of the parallel planes WY, VS, with the plane ED, are parallel (Prop. 12). V W Σ D B .. AE: EB:: AG: GD (th. 61). S Again the intersection AC, GF of the parallel planes XZ, YW, with the plane CG, are parallel. ... AG: GD:: CF: FD. .. substituting the second ratio for the first of this proportion in the previous proportion, we have AE: EB::CF: FD. PROP. XVIII. If a straight line be perpendicular to a plane, every plane which passes through it will be at right angles to that plane. Let the straight line PQ be perpendicular to the plane XZ. Through PQ draw any plane PO, intersecting XZ in the line OQW. Then the plane PO is perpendicular to the plane XZ. For, draw RS, in the plane XZ, perpendicular to WQO. Then, since the straight line PQ is Z W P R X perpendicular to the plane XZ, it is perpendicular to the two straight lines RS, OW, which pass through its foot in that plane. But the angle PQR is contained between PQ, QR, which are perpendiculars at the same point to OW, the common intersection of the planes XZ, PO; this , angle, therefore, measures the angle of the two planes (def. 6); hence, since this angle is a right angle, the two planes are perpendicular to each other. Cor. If three straight lines, such as PQ, RS, OW, be perpendicular to each other, each will be perpendicular to the plane of the other two, and the three planes will be perpendicular to one another. PROP. XIX. If two planes be perpendicular to each other, a straight line drawn in one of the planes perpendicular to their common section will be perpendicular to the other plane. Let the plane VO be perpendicular to the plane XZ, and let OW be their common section. In the plane VO draw PQ perpendicular to OW; Then PQ is perpendicular to the plane XZ. For, from the point Q, draw QR in the plane XZ, perpendicular to Z OW. W P R X Then, since the two planes are perpendicular, the angle PQR is a right angle (def. 6). .. The straight line PQ is perpendicular to the straight lines QR, QO, which intersect at its foot in the plane XZ. .. PQ is perpendicular to the plane XZ (Prop. 5). Cor. If the plane VO be perpendicular to the plane XZ, and if from any point in OW, their common intersection, we erect a perpendicular to the plane XZ, that straight line will lie in the plane VO. For if not, then we may draw from the same point a straight line in the plane VO, perpendicular to OW, and this line, by the Prop., will be perpendicular to the plane XZ. Thus we should have two straight lines drawn from the same point in the plane XZ, each of them perpendicular to this plane, which is impossible. PROP. XX. If two planes which cut each other be each of them perpendicular to a third plane, their common section will be perpendicular to the same plane. Let the two planes VO, TW, whose common section is PQ, be both perpendicular to the plane XZ. Then PQ is perpendicular to the plane XZ. For, from the point Q, erect a perpendicular to the plane XZ. Then, by cor. to last Prop., this straight line must be situated at once Ꮓ V W R X in the planes VO and TW, and is .. their common section. EXERCISES. 1. Prove that but one plane can be passed through a given point perpendicular to a given line. 2. Prove that but one perpendicular can be drawn from a given point to a given plane. 3. That when a plane is perpendicular at the middle of a given line, every point of the plane is equally distant from the extremities of the line, and that every point out of the plane is unequally distant. 4. That through a given line in a plane only one plane perpendicular to the given plane can be passed. 5. That through a line parallel to a given plane but one plane can be passed perpendicular to the given plane. 6. That if two planes which intersect contain two lines parallel to each other, the intersection of the planes will be parallel to the lines. 7. That if a line be parallel to a plane, every other plane passed through this line and meeting the former, will intersect it in a second line parallel to the first. 8. That when a line is parallel to one plane and perpendicular to another, the two planes are perpendicular to each other. 9. That a line parallel to a plane is every where equally distant from that plane. The same of two parallel planes. 10. That two lines are always either in one and the same plane or two parallel planes. Note. These planes, the system of which is unique for each system of two lines not situated in the same plane, are called the parallel planes of these lines. 11. Show that but one plane can be drawn through a given point parallel to a given plane. 12. Prove that two planes parallel to a third are parallel to each other. 13. Draw a perpendicular to two lines not in the same plane. 14. Prove that if two lines are parallel in space, and planes be passed through them perpendicular to a third plane, the two planes will be parallel. 15. That if a line be parallel to one of two perpendicular planes, and a plane be passed through the line perpendicular to the other plane, it will be parallel to the first plane. 16. To place a perpendicular to a given plane at a given point of the plane. 17. To place a plane perpendicular to a given plane, and intersect. ing it in a given line. 18. To place a plane parallel to a given plane. 19. To place a line under a given angle to a given plane. 20. To place a plane under a given angle to a given plane, and intersecting it in a given line. 21. To place a plane perpendicular to two give planes. |