579. Theorem. If two planes are parallel to a third plane, they are parallel to each other. Proof similar to that of § 124. 580. Theorem. If one of two parallel lines is perpendicular to a plane, the other is also; and conversely, two straight lines perpendicular to the same plane are parallel. Given AB || CD, and AB perpendicular to plane P. To prove CD 1 P. A C Proof. Through D draw any line DF E F in plane P. Through B draw BE in plane P and parallel to DF. Then ZABE=ZCDF. ZABE is a rt.Z. § 567 Why? Therefore CDF is a rt. and CD 1 DF, any line in plane P But Outline of proof. Through D draw ED | AB. Then ED I P, and therefore coincides. with CD by § 575. 581. Theorem. distance apart. Two parallel planes are everywhere the same Choose any two points in one plane and draw perpendiculars to the other plane, § 574. Then these lines are perpendicular to the first plane, § 578. Further these two lines are parallel, § 580, and determine a plane intersecting the two parallel planes in parallel lines, § 562. Then apply § 191. EXERCISES 1. Prove the theorem of § 553 by § 580. 2. A line cannot be perpendicular to each of two intersecting planes. PROJECTIONS 582. Definitions. The projection of a point upon a plane is the foot of the perpendicular from the point to the plane. The projection of a line upon a plane is the locus of the projections of all points of the line upon the plane. Thus B is the projection of point A upon plane Q, and EF is the projection of line CD. A D BE F 583. Theorem. The projection upon a plane of a straight line that is not perpendicular to the plane is a straight line. Given AB, a straight line not plane Q. A To prove that the projection of AB upon Q is a straight line. Proof. From any point in AB, as E, draw EF Q. § 574 E G P B HD Plane P, determined by AB and EF, intersects Q in CD. § 551 From G, any other point in AB, draw GHQ. Then GHEF and lies in P. Therefore its foot H lies in CD. And the projections of all points of AB lie in CD. § 574 Why? Why? Furthermore, a perpendicular to Q at any point in CD will intersect AB. Why? Then every point in CD is the projection of a point in AB. .. the projection of AB upon Q is a straight line. 584. Theorem. Of all oblique lines drawn from a point to a plane: (1) Those that have equal projections are equal. (2) Those that have unequal projections are unequal, and the one having the greater projection is the greater. Another statement of this theorem is: Of all oblique lines, drawn to a plane from a point in a perpendicular to the plane, those that cut off equal distances from the foot of the perpendicular are equal, and of those that cut off unequal distances from the foot of the perpendicular, the more remote is the greater. 585. Theorem. The acute angle that a straight line makes with its projection upon a plane is the least angle that it makes with any line passing through its foot and lying in the plane. Given the straight line AB meeting plane Q at B, its projection CB in Q, and DB, any other line through B and in Q. To prove ZABC <ZABD. Proof. and AD. Make BD=BC, and draw AC B Q Then ZABC <ZABD. § 259 586. Definition. The acute angle that a straight line makes with its own projection upon a plane is called the inclination of the line to the plane, or the angle that the line makes with the plane. EXERCISES 1. State and prove the converse theorems to § 584. 2. How does the length of the projection of a line upon a plane compare with the length of the line when: (1) the line is parallel to the plane; (2) the line is perpendicular to the plane; (3) the line is neither parallel nor perpendicular to the plane? 3. Which is the greatest angle that an oblique line to a plane makes with any line in the plane and through its foot? 4. Parallel lines make equal angles with a plane. 5. If a straight line intersects two parallel planes it makes equal angles with the parallel planes. 6. Find the projection of a line 16 in. long upon a plane if the angle it makes with the plane is 45°. If 30°. If 60°. 7. What is the locus of a point in a given plane and equidistant from two given points not in the given plane? 8. A rectangle 8 in. by 12 in. is intersected along one of its diagonals by a plane. If the other diagonal makes an angle of 45° with the plane, find the distance from the extremities of this diagonal to the plane. Ans. 5.098 in. 9. The projection of a given line on each of two planes is a in length. Are the two planes necessarily parallel? Illustrate. DIEDRAL ANGLES 587. Definitions. An angle formed by two intersecting planes is called a diedral angle. The planes are called the faces of the diedral angle, and their intersection is called the edge. If a plane Q revolves about an axis AB, and if a line CD 1 AB is taken in Q, then the plane Q generates a diedral angle, and the line CD generates a plane angle since it revolves in a plane. When the plane angle is acute, right, obtuse, etc., the diedral angle generated in connection K B D with it is acute, right, obtuse, etc. That is, the diedral angle and the plane angle are each formed by the same amount of turning. 588. Plane Angle. The plane angle generated by CD is called the plane angle of the diedral angle. It is formed by two lines, one in each face of the diedral angle, and perpendicular to the edge at the same point. A diedral angle is read by naming its faces, as the diedral angle QP; or it may be read C-AB-D; or simply AB. The words complementary, supplementary, adjacent, vertical, etc., when applied to diedral angles, have meanings corresponding to their meanings when applied to plane angles. 589. The following facts are readily deduced from the definitions: B P Q A (1) All plane angles of a diedral angle are equal. (2) If two diedral angles are equal, their plane angles are equal; and conversely. (3) Two diedral angles are proportional to their plane angles; that is, the plane angle of a diedral angle can be taken as the measure of the diedral angle, EXERCISES 1. By comparison with the corresponding terms in plane geometry, form definitions applying to diedral angles for right, oblique, vertical, adjacent, supplementary, alternate interior. How many of these can be illustrated by the leaves of an open book? 2. With reference to planes, what corresponds to the following axiom in plane geometry: Through a given point only one straight line can be drawn parallel to another straight line? 3. If one plane intersects another plane, the vertical diedral angles are equal. 4. If the sum of two adjacent diedral angles is two right diedral angles, the exterior faces lie in the same plane. 5. If two planes are cut by a third plane so as to make the alternate diedral angles equal, the two planes are 590. Perpendicular Planes. If two planes form a right diedral angle, the two planes are said to be perpendicular to each other. 591. Theorem. If a line is perpendicular to a plane, every plane containing this line is perpendicular to the given plane. Given CD perpendicular to plane PQ, and plane MN through CD intersecting PQ in AB. To prove that plane MN is perpendic Hence diedral ZM-AB-Q is a rt. diedral Z. .. plane MN is perpendicular to plane PQ. Why? § 589 (3) Why? |