MISCELLANEOUS EXERCISES

1. From a point A in one of two intersecting planes, AB is drawn perpendicular to the first plane, meeting the second plane at B, and AC is drawn perpendicular to the second plane, meeting it at C. Show that BC is perpendicular to the line of intersection of the two planes.

2. Two line segments AB and DC are such that if BC and AD are joined each of the angles A, B, C, D, is a right angle. Prove that AB and DC are parallel.

3. If two triangles ABC and A'B'C' in different planes are so situated that the lines AA', BB', CC' meet in a point S, then the pairs of corresponding sides AB, A'B'; AC, A'C'; BC, B'C' intersect in points of one straight line.

SUGGESTION. Let the planes of ABC and A'B'C' intersect in a straight line s. The points S, A, A', B, B' lie in one plane intersecting s at X, AB and A'B' must intersect at X. Similarly for the other pairs.

say.

4. Show that the locus of a point equidistant from three given points is a straight line through the centre of the circle determined by the three points.

5. Prove that if AP makes equal angles with AB and AC, it must lie in one or other of two fixed planes.

6. Of three given planes each is perpendicular to the other two. Show that of their three lines of intersection each is perpendicular to the other two.

7. If two intersecting planes are cut by two parallel planes, the lines of section of the first two by each of the parallel planes will make equal angles.

8. If a straight line is equally inclined to two given planes, show that it is perpendicular to the plane bisecting one of the dihedral angles formed by the two planes.

9. Show that the locus of the foot of the perpendiculars drawn from a fixed point to the planes through a fixed straight line is a circle.

10. Draw through the vertex of a trihedral angle a straight line which will make equal angles with the three edges of the trihedral angle. many such lines are there ?

How

11. The projections on a plane of two equal and parallel line-segments are equal and parallel.

1. DEFINITIONS.

SUMMARY OF CHAPTER VI

(1) Solid Figure-one whose points and lines do not all lie in the same plane. § 386.

(2) Line Parallel to a Plane -a straight line which does not meet the plane, however far they may be extended. § 392. (3) Line Perpendicular to a Plane · a straight line which is perpendicular to every line of the plane passing through their intersection. § 393.

a straight line neither parallel nor

perpendicular to the plane. § 393.

(5) Parallel Planes· planes which do not meet, however far they may be extended. § 393.

(6) Skew or Gauche Lines-straight lines so situated that no plane can contain them both. Ex. 3, p. 271.

(7) Distance from a Point to a Plane

dicular from the point to the plane.

the length of the perpen§ 411.

(8) Dihedral Angle - the figure formed by two planes meeting in a straight line. § 426.

(9) Right Dihedral Angle ·

an angle formed by two intersecting planes when the adjacent angles so formed are equal. § 429. (10) Plane Angle of a Dihedral Angle - the angle between two straight lines drawn perpendicular to the edge from the same point, one in each boundary. § 430.

(11) Projection of a Point upon a Plane · the foot of the perpendicular drawn from the point to the plane. § 445.

(12) Projection of a Line upon a Plane - the locus of the projections of its points upon the plane. § 445.

(13) Inclination of a Line to a Plane - the angle between the line and its projection upon the plane. § 447.

(14) Trihedral Angle - the figure formed by three planes meeting at a point. § 453.

(15) Polyhedral Angle·

at a point. § 453.

the figure formed by several planes meeting

(16) Identically Equal Polyhedral Angles — two which can be made to coincide. § 455.

(17) Symmetrical Polyhedral Angles-two whose parts are equal, respectively, but arranged in reverse order. § 456.

(18) Isosceles Trihedral Angle - one having two face angles equal.

§ 462.

2. POSTULATE.

(1) Through three points not in the same straight line there can pass one and only one plane (Postulate 9). § 388.

3. ELEMENTS WHICH DETERMINE A PLANE.

(1) Three points (Postulate 9). § 388.

(2) A straight line and a point not lying on it. § 388.

(3) Two intersecting straight lines. § 388.

(4) Two parallel straight lines. § 388.

4. PROBLEMS.

(1) At a given point in a given plane to erect a perpendicular to the plane. § 403.

(2) From a given point without a given plane to draw a perpendicular to the plane. § 408.

(3) To draw a straight line which shall be perpendicular to each of two given straight lines not lying in the same plane. § 450.

5. THEOREMS ON THE INTERSECTIONS OF PLANES.

(1) The intersection of two planes is a straight line. § 390.

(2) If two planes have one point in common, they must have a second point, and hence a straight line, in common. § 391.

(3) If three planes intersect, two and two, their three lines of intersection are either concurrent, or are parallel, two and two. § 394.

(4) The two planes determined by two given parallel lines and a point not lying in their plane, intersect in a line parallel to each of the given lines. § 395.

(5) Two parallel planes are intersected by any third plane in parallel lines. § 419.

6. THEOREMS ON STRAIGHT LINES PERPENDICULAR TO PLANES. (1) If a straight line is perpendicular to each of two given straight lines at their point of intersection, it is perpendicular to the plane of these lines. § 397.

(2) At any point of a straight line one plane can be constructed perpendicular to that line, and only one. § 398.

(3) Through a given point not on a given straight line, one plane and only one can be constructed perpendicular to the given line. § 399.

(4) Two intersecting planes cannot both be perpendicular to the same straight line. § 400.

(5) All the straight lines perpendicular to a given line at a given
point lie in one plane perpendicular to the given line. § 401.
(6) If a plane bisects a given line-segment perpendicularly, every
point of the plane is equidistant from the extremities of the
line-segment, and conversely. § 402.

(7) At a point in a plane but one straight line can be drawn perpen-
dicular to the plane. § 404.

(8) From a point without a plane only one perpendicular to the plane can be drawn. § 409.

(9) Two straight lines perpendicular to the same plane are parallel.

§ 405.

(10) If one of two parallel lines is perpendicular to a plane, the other is also. § 407.

(11) Two planes perpendicular to the same straight line are parallel.

§ 412.

(12) A straight line perpendicular to one of two parallel planes is also perpendicular to the other. § 422.

(13) If a straight line is perpendicular to a given plane, every plane containing that line is perpendicular to the given plane. § 436.

7. THEOREMS ON STRAIGHT LINES AND PLANES PARALLEL TO THEM. (1) If two straight lines are parallel, any plane containing one of them, and not the other, is parallel to the other. § 413.

(2) Through either of two given straight lines not lying in the same plane, one plane can be passed parallel to the other line. § 414. (3) Through a given point a plane can be passed parallel to any two given straight lines in space. § 415.

(4) If a straight line is parallel to a given plane, it is parallel to the intersection of any plane through it, with the given plane. § 416.

(5) If a straight line is parallel to a given plane, a line drawn from any point in the plane parallel to the given line lies in the given plane. § 417.

(6) If two intersecting straight lines are each parallel to a given plane, the plane determined by these lines is also parallel to the given plane. § 418.

8. THEOREMS ON PLANES PERPENDICULAR TO EACH OTHER.

(1) If a straight line is perpendicular to a given plane, every plane containing that line is perpendicular to the given plane. § 436. (2) Any plane perpendicular to the edge of a dihedral angle is perpendicular to each of its faces. § 438.

(3) If two planes are perpendicular to each other, a straight line drawn in one of them, perpendicular to their intersection, is perpendicular to the other. § 439.

(4) If two planes are perpendicular to each other, a straight line drawn from any point of their intersection, perpendicular to one plane, must lie in the other. § 440.

(5) If two planes are perpendicular to each other, a straight line drawn from any point in one of them, perpendicular to the other, must lie in the first plane. § 442.

(6) If two intersecting planes are each perpendicular to a third plane, their intersection is also perpendicular to that plane. § 443.

9. THEOREMS ON PARALLEL PLANES.

(1) Two parallel planes are intersected by any third plane in parallel lines. § 419.

(2) Parallel line-segments terminated by parallel planes are equal.

§ 420.

(3) Two parallel planes are everywhere equidistant. § 421.

(4) A straight line perpendicular to one of two parallel planes is also perpendicular to the other. § 422.

(5) If two straight lines are cut by three parallel planes, the corresponding segments are proportional. § 425.

10. THEOREMS ON DIHEDRAL ANGLES.

(1) All plane angles of the same dihedral angle are equal. § 431.
(2) The lines of intersection with the boundaries of a dihedral angle,
of any plane perpendicular to the edge of that angle, form the
plane angle of the dihedral angle. § 432.

(3) Two dihedral angles are equal if their plane angles are equal. § 433.

(4) Two dihedral angles are in the same ratio as their plane angles.

§ 434.

(5) The locus of points equidistant from the boundaries of a dihedral angle is the plane bisecting that angle. § 444.

11. THEOREMS ON TRIHEDRAL AND POLYHEDRAL ANGLES.

(1) The sum of any two face angles of a trihedral angle is greater than the third face angle. § 458.

(2) Any face angle of a polyhedral angle is less than the sum of the remaining face angles. § 459.

(3) The sum of the face angles of any convex polyhedral angle is less

than four right angles. § 460.