Related to this proposition is the problem of supporting a tall iron smokestack by wire stays. Evidently three stays are needed, and they are preferably placed at the vertices of an equilateral triangle, the smokestack being in the center. The practical problem may be given of locating the vertices of the triangle and of finding the length of each stay. THEOREM. _Two straight lines perpendicular to the same plane are parallel._ Here again we may cut through the figure by the plane of the two parallels, and we get the figure of plane geometry relating to lines that are perpendicular to the same line. The proposition shows that the opposite corners of a room are parallel, and that therefore they lie in the same plane, or are _coplanar_, as is said in higher geometry. It is interesting to a class to have attention called to the corollary that if two straight lines are parallel to a third straight line, they are parallel to each other; and to have the question asked why it is necessary to prove this when the same thing was proved in plane geometry. In case the reason is not clear, let some student try to apply the proof used in plane geometry. THEOREM. _Two planes perpendicular to the same straight line are parallel._ Besides calling attention to the corresponding proposition of plane geometry, it is well now to speak of the fact that in propositions involving planes and lines we may often interchange these words. For example, using "line" for "straight line," for brevity, we have: One _line_ does not determine One _plane_ does not determine a _plane_. a _line_. Two intersecting _lines_ Two intersecting _planes_ determine determine a _plane_. a _line_. Two _lines_ perpendicular to Two _planes_ perpendicular to a _plane_ are parallel. a _line_ are parallel. If one of two parallel _lines_ If one of two parallel _planes_ is perpendicular to a _plane_, the is perpendicular to a _line_, the other is also perpendicular to other is also perpendicular to the _plane_. the _line_. If two _lines_ are parallel, every If two _planes_ are parallel, _plane_ containing one of the every _line_ in one of the _planes_ _lines_ is parallel to the other is parallel to the other _plane_. _line_. THEOREM. _The intersections of two parallel planes by a third plane are parallel lines._ Thus one of the edges of a box is parallel to the next succeeding edge if the opposite faces are parallel, and in sawing diagonally through an ordinary board (with rectangular cross section) the section is a parallelogram. THEOREM. _A straight line perpendicular to one of two parallel planes is perpendicular to the other also._ Notice (1) the corresponding proposition in plane geometry; (2) the proposition that results from interchanging "plane" and (straight) "line."