SOLID GEOMETRY. PLANES. DEFINITION. A solid has length, breadth, and thickness. PROPOSITION I. One part of a straight line cannot be in a plane, and another part out of it. P For if it be possible, let one part AB of the straight line ABC be in the plane PN, and another part BC out of it. In the plane PN, produce AB in a straight line to X. Let a plane through ABX be turned about ABX till it passes through C, and .. contains BC. Then the two straight lines ABC, ABX in one plane have a common segment; which is impossible. ... One part of a straight line, &c. PROPOSITION II. Two straight lines which cut one another are in one plane. Let the two straight lines AB, CF cut one another in G. Then shall AB, CF lie in one plane. For let a plane through AB be turned about AB till it passes through C. Then since this plane contains C and G, it contains the straight line CGF. Hence AB, CF lie in this plane. PROPOSITION III. Three straight lines which cut one another and form a triangle are in one plane. B Let the three straight lines AB, BC, CA cut one another and form a A. Then shall AB, BC, CA lie in one plane. For let a plane containing BC be turned about BC till it passes through A. Then since this plane contains A and B it contains AB, and since it contains C and A it contains CA. Hence AB, BC, CA lie in this plane. PROPOSITION IV. If two straight lines are parallel, the straight line drawn from any point in the one to any point in the other is in the same plane with the parallels. Let AB, CF be parallel straight lines, P, Q any points in AB, CF respectively. Then shall the straight line joining P, Q be in the same plane as AB, CF. For if not, if possible, let it be out of this plane as PXQ, and in this plane draw a straight line PNQ joining P and Q, then there are two straight lines PNQ, PXQ inclosing a space; which is impossible. Hence, If two straight lines, &c. PROPOSITION V. If two planes cut one another, their common section is a straight line. Let PIQ be the common section of two planes AB, CD which cut one another; PIQ shall be a straight line. For if not, if possible, let a straight line PXQ be drawn from P to Q in the plane AB, and a straight line PYQ from P to Q in the plane CD. Then there are two straight lines PXQ, PYQ inclosing a space; which is impossible. .. If two planes cut, &c. DEFINITION. A straight line is said to be perpendicular to a plane when it is perpendicular to every straight line in that plane which meets it. PROPOSITION VI. If a straight line be perpendicular to each of two straight lines at their point of intersection, it shall be perpendicular to the plane passing through them. Let PQ be to QA and QC. Then shall PQ be to the plane passing through QA, QC. Through in the same plane as QA, QC draw any other straight line QH; and through any point H in QH draw a straight line cutting Q4, QC in A and C; produce PQ making QR = PQ; join PA, PC, PH, RA, RC, RH. == Then PQ, QA and right, PQA are respectively = RQ, QA and RQA; .. PA RA. (I. I) Similarly PC = RC, and AC is common to as PCA, RCA ; .. LPCA = LRCA; (1.5) also PC, CH are respectively = RC, CH; also PQ, QH are respectively = RQ, QH; .. PQ is to the plane through QA, QC. (Def.) (1.5) |