In the figure let A C, D F be any two lines cut by three planes, RS, P Q, M N, in the points A, B, C, D, E, F. Compare the segments of the lines. Sug. Join A, F. and F A and FD. Pass a plane through A C and A F, (Why can we do this?) Let C F and B G be the intersections made by the first plane with the others and A D and G E the intersections made by the second Cor. I. Suppose two lines are cut by any number of || planes. What can you say or prove of the corresponding segments? 417. Cor. II. If x straight lines be cut by three || planes, prove how the corresponding segments are related. Given: P, a point without the plane M N, and P O, a perpendicular to it. 1. Compare PO with any other line drawn from P to the plane. In how many ways can you make the comparison? Generalize. ≤ = = A (angles 2. Given: POL M N and B of inclination), to compare P B and P A. converse and prove it. Generalize. 3. If the projections of two lines from the same point to the same plane are equal, how are the lines related? State and prove the converse. Generalize each. 4. Given the unequal oblique 4s P BO > PCO, to compare P B and P C. State and prove the converse. Generalize each. 5. Suppose the projections of two oblique lines drawn from a point, P, to the plane M N to be unequal, compare the lines projected. Write a general statement comprehending the five statements above. Begin in this manner: Of all lines that can be drawn from a point to a plane 1. ——————, 2. 3. 5. EXERCISES. 4. 405. Parallel line segments are proportional to their projections on a plane. 406. Can you project two lines on three different planes at the same time. 419. PROPOSITION XIV. W R P S Let W P be any line intersecting the plane R S at point P, and let O P be its projection on R S. Let Q T be a line in RSLOP. Show how QT is related to the given line W P. Sug. Measure off on Q T, PH=PI. Join O and H, O and I, W and H, W and I. Compare As POI and P OH, also As IP W and HP W. How is W P Generalize the truth reached? I W and H W; related to Q T? EXERCISES. 407. If a line is || to each of two intersecting planes, how is it related to their intersection? Prove. 408 Can you construct a plane containing a given line and to another line? 409. If two lines intersect the same plane, show that they are equally inclined to it. DIEDRAL ANGLES. 420. Definition: When any number of planes pass through the same line, they are said to form a pencil of planes, and any two of the planes form a diedral angle. M R 421. Q The planes are the faces of the diedral angle, and their intersection the edge. 422. A diedral angle may be designated by two letters on its edge, but if several diedral angles have a common edge, then four letters are necessary, one in each face and two on the edge, thus: SDCP, NDC M. 423. Definition: The plane angle of a diedral is the angle formed by two straight lines, one in each plane, drawn perpendicular to the edge at any given point. Thus if B A and C A in the faces D F and E G, respectively, are each 1D E, they form the plane of the diedral D E. 424. By using your cardboard and by drawings, illustrate vertical diedral angles, adjacent diedral angles, right diedral angles. Write a definition of each. [Note. The faces of the diedral angle are indefinite in extent, but for convenience in study we take a limited portion of the bounding planes. The pupil should take pains in learning to draw the figures in Solid Geometry.] 425. What plane angle will be formed if a plane be passed perpendicularly to the edge of a diedral angle and intersecting its sides? Through a given line in a plane, how many perpendicular planes can be passed to the given plane? Why? |