3D ANGLES by Shaun Netherby

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3D ANGLESby Shaun Netherby

• Tri-dimensional trickery and multi-dimensional
  madness featuring dihedral and trihedral angles!

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Dihedral Angles

• Definition A dihedral angle is a figure in space
  formed by two half planes
• and that intersect in a
  common boundary line l.

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The Plane Angle

• Definition A plane angle is the angle formed by
  taking two points equidistant from a common point
  on l on the two half planes.

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Theorems and Lemmas oh my!
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Proof

• We may assume the marked congruencies are true.
  Since sides AB and AB are parallel and
  congruent, AABB is a parallelogram. So BB
  AA and . Similarly CC AA and
  , so and BBCC, so
  BCCB is a parallelogram. Thus BC BC.
  Since AB AB and AC AC, so ?ABC ?ABC
  by SSS. Therefore, .

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lemma
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Proof

• Consider the plane angle BAC. Suppose A is any
  other point on l and B and C are chosen in
  and ,
  respectively, such that l and l
  

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Proof (cont)

• We have l (A,B,C) and l (A,B,C),
  so
• (A,B,C) (A,B,C). Because
  is a cutting plane of (A,B,C) and
  (A,B,C), . Similarly, we can show
  that . Using the preceding lemma, it
  immediately follows that

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Trihedral Angles

• A trihedral angle is formed when the edges of
  three half-planes meet at a single point, the
  vertex.
• The angle consists of the vertex, the three edges
  and and the faces composed of
  the half planes, bounded by the edges. The three
  angles are the face angles.

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Theorem

• The sum of any two face angles of a trihedral
  angle is greater than the third face angle.
• Proof see book page 164-165.

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Theorem

• The sum of the face angles of any trihedral angle
  is less than 360

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Proof

• Consider a tetrahedron the figure bounding the
  three face edges of a trihedral angle. Noting
  the four trihedral angles and the previous
  theorem, notice

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Proof (cont)

• Adding the three inequalities we get
• which is,
• resulting in our proof,

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The 6 congruence factors for congruent trihedral
angles

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Theorem

• If the three face angles of one trihedral angle
  are congruent respectively to the three face
  angles of another trihedral angle, then the two
  trihedral angles are either congruent or
  symmetric.

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One more theorem? Why not

• Given trihedral angles and
• such that the dihedral angle at is
  congruent to the dihedral angle at ,
  and
• . Then the two trihedral angles
  are either congruent or symmetric.

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Congruence / Symmetry

• Original
• Congruent Symmetric

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Homework

• Found nicely written in the book, 12.19, p.170
• If angle AB1B2B3B4 is a quadrehedral angle, prove
  that angB1AB2 angB2AB3 angB3AB4 angB4AB1 lt
  360