Plane and solid geometry . Only 1. In plane MKj through C, draw any line CD, and let theplane determined hj AC and CD intersect plane RS in EF, 2. Then CD II EF. 3. But AB ± EF, 4. .-. AB J_ CD, any line in plane MN passing through C, 5. .. line AB ± plane MN. 652. Cor. I. Through a given point there exists onlyone plane parallel to a given plane, (Hint. Apply §§ 638,6445, 651, 625.) 653. §§ 6445 and 652 may be combined in one statement:Through a given point there exists one and onhj one plane parallel to a given plane. 654. Cor. II. If two planes are each parallel to a thirdplane, they
Plane and solid geometry . Only 1. In plane MKj through C, draw any line CD, and let theplane determined hj AC and CD intersect plane RS in EF, 2. Then CD II EF. 3. But AB ± EF, 4. .-. AB J_ CD, any line in plane MN passing through C, 5. .. line AB ± plane MN. 652. Cor. I. Through a given point there exists onlyone plane parallel to a given plane, (Hint. Apply §§ 638,6445, 651, 625.) 653. §§ 6445 and 652 may be combined in one statement:Through a given point there exists one and onhj one plane parallel to a given plane. 654. Cor. II. If two planes are each parallel to a thirdplane, they are parallel to each other. (Hint. See § 180.) 655. Def. The projection of a point upon a plane is the footof the perpendicular from the point to the plane. 656. Def. The projection of a line upon a plane is the locusof the projections of all points of the line upon the plane. BOOK VI 319 Proposition XVIII. Theorem 657. The projection upon a plane of a straight linenot perpendicular to the plane is a straight Given str. line AB not J_ plane MN. To prove the projection of AB upon MN a str. line. Argument 1. Through (7, any point in AB^ draw CD JL plane MN. 2. Let the plane determined by AB and CD intersect plane MN in the str. line EF, 3. From H, any point in AB^ draw HK^ in plane AF, II CD. 4. Then HK A. plane MN 6. ,\ K is the projection of H upon planeMN. 6. .*. EF is the projection of AB upon plane MN. 7. .. the projection of AB upon plane MN is a str. line. Reasons 1. § 639. 2. §§ 612, 616. 3. § 179. 4. § 636. 5. § ^55. 6. § 656. 7. Args. 2 and 6. Ex. 1177. Compare the length of the projection of a line upon aplane with the length of the line itself :(a) If the line is parallel to the plane. (6) If the line is neither parallel nor perpendicular to the plane,(c) If the line is perpendicular to the plane. 320 SOLID GEOMETRY Propositiox XIX. Theorem 658. Of all oblique lines drawn from a point to a -plane: I. Those having equal projections are equa
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Keywords: ., bookcentury1900, bookdecade1910, booksubjectgeometr, bookyear1912
