Plane and solid geometry . # ^ / ? ^ / • ^ / « \ / t % / • ^ i ^ ;-> \ % The proofs are left as exercises for the student. Hint. Let V and F be the two trihedral A with parts equal butarranged in reverse order. Construct trihedral Z V symmetrical to what will be the relation of T^ to T^ ? of F to T? Ex. 1240. Can two polyhedral angles be symmetrical and equal ?vertical and equal ? symmetrical and vertical ? If two polyhedral anglesare vertical, are they necessarily symmetrical ? if symmetrical, are theynecessarily vertical ? Ex. 1241. Are two trirectangular trihedral angles necessari


Plane and solid geometry . # ^ / ? ^ / • ^ / « \ / t % / • ^ i ^ ;-> \ % The proofs are left as exercises for the student. Hint. Let V and F be the two trihedral A with parts equal butarranged in reverse order. Construct trihedral Z V symmetrical to what will be the relation of T^ to T^ ? of F to T? Ex. 1240. Can two polyhedral angles be symmetrical and equal ?vertical and equal ? symmetrical and vertical ? If two polyhedral anglesare vertical, are they necessarily symmetrical ? if symmetrical, are theynecessarily vertical ? Ex. 1241. Are two trirectangular trihedral angles necessarily equal?Are two Mrectangular trihedral angles equal ? Prove your answers. Ex. 1242. If two trihedral angles have three face angles of one equalrespectively to three face angles of the other, the dihedral angles of thefirst are equal respectively to the dihedral angles of the second. BOOK VI 341 Proposition XXXI. Theorem 710. The suin of any two face angles of a trihedralangle is greater than the third face angle- V. Given trihedral Z. V-ABC in whicli the greatest faceZ is .4 VB. To prove Z BVC + Z CVA > Z AVB, Outline of Proof 1. In face AVB draw VD making Z DVB= Z BVC, andthrough D draw any line intersecting VA in E and VB in F, 2. On VC lay off VG = VD and draw FG and GE. 3. Prove A FVG = ADVF-, then FG = FD, 4. But FG + GE > FD -\~ DE, .: GE > DE. 5. In A GVE and EVD, prove Z GVE > Z EVD. 6. ^wi Zfvg = Zdvf. 7. ,\ Z FVG-{-ZGVE> ZeVD -\-ZDVF^ Zbvc + Z CVA -> Zavb. 711. Question. State the theorem in Bk. I that corresponds toProp. XXXI. Can that theorem be proved by a method similar to theone used here ? If so, give the proof. Ex. 1243. If, in trihedral angle V-ABC, angle BVC = 60^, and angleCVA = 80°, make a statement as to the number of degrees in angle AVB. Ex. 1244. Any face angle of a trihedral angle is greater than thedifference of the other two. 342 SOLID GEOMETRY PpvOpositiox XXXII. Theorem 712. The sum of all tlve face angles of an


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Keywords: ., bookcentury1900, bookdecade1910, booksubjectgeometr, bookyear1912