Section 2. The Relative Position of a Line Theorem 6. If a line is perpendicular to each of two intersecting lines, it is perpendicular to every other line lying in their plane and passing through their point of intersection. Given x and z, two lines inter secting at O, and w perpendicular To prove = P'O; let any transversal cut M P and P' with A, B, C. 4. B I, th. 20, cor. 5 ACP'. I, th. 12 .. by folding A ACP over AC as an axis, it can be brought to coincide with A ACP.' 5. ..A BOPA BOP'. 6... POB is a rt. , and wy. Def. congruence I, th. 12 Why? DEFINITIONS. A line is said to be perpendicular to a plane when it is perpendicular to every line in that plane which passes through its foot, that is the point where it meets the plane. The plane is then said to be perpendicular to the line. If a line meets a plane, and is not perpendicular to it, it is said to be oblique to the plane. COROLLARY. If a line is perpendicular to each of two intersecting lines it is perpendicular to their plane. Theorem 7. If a line is perpendicular to each of three concurrent lines at their point of concurrence, the three lines are coplanar. 5. .. it is absurd to suppose OC not in M with OA and OB. COROLLARIES. 1. Lines perpendicular to the same line at the same point are coplanar. 2. Through a given point in a plane there cannot be drawn more than one line perpendicular to that plane. Suppose OP and OQ plane M. Then each would be perpendicular to OX, the line of intersection of their plane N with the given plane M, thus violating Prel. th. 2. 3. Through a given point in a line there cannot be drawn more than one plane perpendicular to that line. M N Q For if two planes could be drawn perpendicular to the line, then three lines in each would be perpendicular to the given line, hence the two planes would coincide. EXERCISE. 580. Prove that if the hand of a clock is perpendicular to its moving axle, it describes a plane in its revolution. 7. draw OZ, ZY, XY. ... XZ OY, OX OX, = <XOY = OXZrt., .. Δ ΧΟΥΞ Δ ΟΧΙ, OZ = XY. OY; A X ...ZY ZY, ... A XYZA OZY, ZYXZ = ZOY = rt. Z. .. XA, XY, XO are coplanar. ... YO lies in that same plane. Z I, th. 1 I, th. 12 Why? Post. 7 But YO and AX OX. Def.to a plane 8... YO II AX, and similarly for all other 1. I, th. 16, cor. 3 COROLLARY. From a point outside of a plane, not more than one line can be drawn perpendicular to that plane. EXERCISES. steps 1-6 ? 581. Why would not steps 7, 8 be sufficient, without 582. Are lines which make equal angles with a given line always parallel? (Answer by drawing figures to illustrate.) 583. Show how to determine the perpendicular to a plane, through a given point, by the use of two carpenter's squares. 584. Prove th. 6 on the following outline: Assume B on y, and draw ABC so that AB = BC (How is this done?); prove 2 PB2 + 2 · BC2 = PA2+ PC22 PO2 + OC2 + OA22 PO2 + 2 OB2 + 2 BC2; .. = PO2 + OB2; .. POB is a rt. Z. Theorem 9. If one of two parallel lines is perpendicular to a plane the other is also. Def.plane 3. But YOA = ▲ Y'O'A', ≤ YOB = ZY'O'B'. 4. .. Y'O'A', Y'O'B' are also rt. . MN. Prel. th. 1 Def. plane DEFINITIONS. The projection of a point on a plane is the foot of the perpendicular through that point to the plane. The projection of a line on a plane is the locus of the projections of all of its points. Theorem 10. The projection of a straight line on a plane is the straight line which passes through the projections of any two of its points. Given A', P', B', the projec tions of A, P, B, points in the line AB, on the plane MN. N A' COROLLARY. If a line intersects a plane, its projection passes through the point of intersection. Th. 1, cor. 2 Th. 1, cor. 3 Th. 2 DEFINITIONS. The smallest angle formed by a line and its projection on a plane is called the inclination of the line to the plane or the angle of the line and the plane. A figure is said to be projected on a plane when all of its points are projected on the plane. The plane determined by a line and its projection on another plane is called the projecting plane. In the figure of th. 10, B'OB is the inclination of AB to MN. The plane determined by AB, A'B' is the projecting plane. Theorem 11. Of all lines that can be drawn from a point to a plane, 1. The perpendicular is the shortest; 2. Obliques with equal inclinations are equal, and conversely; 3. Obliques with equal projections are equal, and conversely; 4. Of two obliques with unequal inclinations, that having the greater inclination is the shorter, and conversely; 5. Of two obliques with unequal projections, that having the longer projection is the longer, and conversely. 1. Given PO plane MN, PX oblique to MN. P N Proof. 1. A POY A POX, and .. PY PX. I, th. 19, cor. 7 For the converse: 2. A POY A POX, and PYO = PXO. Why? ་ |