Elements of geometry and trigonometry . ightbe erected to the same plane, which is impossible (Prop. 2.). Cor. 2. Two lines A and B, parallel to a third C, are par-allel to each other ; for, conceive a plane perpendicular to theline C ; the lines A and B, being parallel to C, will be perpen-dicular to the same plane ; therefore, by the preceding Corol-lary, they will be parallel to each other. The three lines are supposed not to be in the same piane ;otherwise the proposition would be already known (Book XXII.). PROPOSITION VIII. THEOREM. If a straight Une is parallel to a stra
Elements of geometry and trigonometry . ightbe erected to the same plane, which is impossible (Prop. 2.). Cor. 2. Two lines A and B, parallel to a third C, are par-allel to each other ; for, conceive a plane perpendicular to theline C ; the lines A and B, being parallel to C, will be perpen-dicular to the same plane ; therefore, by the preceding Corol-lary, they will be parallel to each other. The three lines are supposed not to be in the same piane ;otherwise the proposition would be already known (Book XXII.). PROPOSITION VIII. THEOREM. If a straight Une is parallel to a straight line drawn in a plane,it will he. parallel to that plane. Let AB be parallel to CDof the plane NM ; then willit be parallel to the planeNM. For, if the line AB, whichlies in the plane ABDC,could meet the plane MN,this could only be in somepoint of the line CD, the common intersection of the twoplanes : but AB cannot meet CD, since they are parallel ;hence it will not meet the plane MN ; hence it is parallel tothat plane (Def. 2.).. PROPOSITION IX. THEOREM. Two planes which are perpendicular to the same straight line,are parallel to each other. BOOK VI. 133 O Let the planes NM, QP, be per- ^l pendicular to the line AB, then willihf • be parallel. ror, it they can meet any where,let O be one of their commonpoints, and draw OA, OB ; the lineAB which is perpendicular to theplane MN, is perpendicular to thestraight line OA drawn through its foot in that plane ; for thesame reason AB is perpendicular to BO ; therefore OA and OBare two perpendiculars let fall from the same point O, uponthe same straight line ; which is impossible (Book 1. Prop. XIV.);therefore the planes MN, PQ, cannot meet each other ; conse-quently they are parallel. ?\ A D\ p T^ \ \ \ B \ PROPOSITION X. THEOREM. If a plam cat two parallel planes, the lines of intersection will be parallel. Let the parallel planes NM,QP, be intersected by the planeEll ; then will the lines of inter-section EF, Gil, be paralle
Size: 2165px × 1154px
Photo credit: © The Reading Room / Alamy / Afripics
License: Licensed
Model Released: No
Keywords: ., bo, bookcentury1800, booksubjectgeometry, booksubjecttrigonometry
