Plane and solid geometry . MNO = Z mno. Reasons1. § 54, 14. 2. § 670. 3. § 62. 4. §§ 670, 62. 5. § 18. 674. Cor. I. The plane angle of a right dihedral angleis a right angle, 675. Cor. II. If two intersecting planes are each per-pendicular to a third plane, their intersections with tJiethird plane intersect each other. BOOK VI 325 Given planes ABand CI) JL plane 3fiVand intersecting eachother in line DB; alsolet AE and FC be theintersections of planesAB and CD with plane J/xV. To prove that AEand FC intersect eachother. Argu:ment 1. Either AE 11 FC or AE and i^C intersect each other. 2.
Plane and solid geometry . MNO = Z mno. Reasons1. § 54, 14. 2. § 670. 3. § 62. 4. §§ 670, 62. 5. § 18. 674. Cor. I. The plane angle of a right dihedral angleis a right angle, 675. Cor. II. If two intersecting planes are each per-pendicular to a third plane, their intersections with tJiethird plane intersect each other. BOOK VI 325 Given planes ABand CI) JL plane 3fiVand intersecting eachother in line DB; alsolet AE and FC be theintersections of planesAB and CD with plane J/xV. To prove that AEand FC intersect eachother. Argu:ment 1. Either AE 11 FC or AE and i^C intersect each other. 2. Suppose J-E: 11 FC. Then through i7, any point in DB, pass a plane ^^Z _L jFC,intersecting FC in iT and ^i5J in Z. 3. Then plane HKL is _L AE also. 4. .. Z ^iTL is the plane Z. of dihedral Z i^C, and Z KLH is the plane Z of di-hedral Z ^jE^. But dihedral A FC and AE are rt. dihe-dral A. ,, A HKL and ^LIT are rt. A, ..A ifiTi contains two rt. A. 8. But this is impossible. 9. .*. AE and i^Cintersect each other, 5. 6. 7. Reasons 1. 161, a. 2. § 627. 3. § 636. 4. § 670. 5. § 672. 6. § 674. 7. Arg. 6. 8. § 206. 9. § 161, b. Ex. 1195. Find the locus of all points equidistant from two givenpoints in space. Ex. 1196. Find the locus of all points equidistant from three givenpoints in space. Ex. 1197. Are the supplements of equal dihedral angles equal ?complements ? Prove your answer. Ex. 1198. If two planes are each perpendicular to a third plane,can they be parallel to each other? Explain. If they are parallel toeach other, prove their intersections with the third plane parallel. 326 SOLID GEOMETRY Proposition XXI. Theorem (Converse of Prop. XX) 676. If the plane angles of two dilvedral angles areequal, the dihedral angles are equal.
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Keywords: ., bookcentury1900, bookdecade1910, booksubjectgeometr, bookyear1912
