Let the two parallel lines A B and C D be included between the parallel planes M N and PQ. We are to prove If AB and CD be A B C D. to the two Il planes they are equal, § 434 (if two planes be ||, all the points of either are equally distant from the other). If A B and CD be not the points A and C the lines A E and C F to the two planes, draw from to the plane M N. § 458 $434 $ 430 (if a straight line be to a plane it is to any line of the plane drawn and through its foot); LBAE LDCF, = $462 (if two not in the same plane have their sides || and lying in the same direction they are equal). PROPOSITION XII. THEOREM. 465. The intersections of two parallel planes by a third plane are parallel lines. Let the plane OS intersect the parallel planes P Q and M N in the lines A C and B D respectively. PROPOSITION. XIII. THEOREM. 466. If a straight line be perpendicular to one of two parallel planes it is perpendicular to the other. Let MN and PQ be parallel planes and AB be per Let two planes embracing AB intersect the planes M N and PQ in AC, BE and A D, B F respectively. Then AC is to BE and AD to BF, § 465 (the intersections of two || planes by a third plane are || lines). But EB and FB are (if a straight line be to a plane it is to every straight line of the plane drawn through its foot). to A B, $430 .. AC and AD which are respectively to BE and BF are to A B, (if a straight line be to one of two || lines, it is to the other). $67 § 449 (if a line be to two straight lines in a plane drawn through its foot it is to the plane). Q. E. D. 467. COROLLARY. If two planes be parallel to a third plane. they are parallel to each other. For, every line perpendicular to this third plane is perpendicular to the other planes; and two planes perpendicular to a straight line are parallel. PROPOSITION XIV. THEOREM. 468. If a straight line be parallel to another straight line drawn in a plane, it is parallel to the plane. Let AC be parallel to the line B D in the plane M N. We are to prove AC to the plane M N. From A and C, any two points in A C, draw A B and C D 1 to A C, and A E and C F 1 to the plane M N. (if two & not in the same plane have their sides || and lying in the same direction, they are equal). Now since the points A and C, any two points in the line A C, are equally distant from the plane MN, all the points in AC are equally distant from the plane MN. .. A C is to the plane MN. $432 Q. E. D. PROPOSITION XV. THEOREM. 469. If two straight lines be intersected by three parallel planes their corresponding segments are proportional. Let A B and C D be intersected by the parallel planes MN, PQ, RS, in the points A, E, B, and C, F, D. (the intersections of two || planes by a third plane are || lines). $465 $ 275 (a line drawn through two sides of a ▲l to the third side divides those Also, sides proportionally). GF is to A C, § 465 $275 Ax. 1. Q. E. D. |