(Geometric Structures)

Angles

prepared by:

Joseph Malkevitch
Department of Mathematics and Computing
York College (CUNY)
Jamaica, New York 11451

Email: malkevitch@york.cuny.edu

web page:

http://www.york.cuny.edu/~malk

1. The diagram below shows a triangle in the Euclidean plane with the sides of the triangle extended at each vertex, so that CA is extended to A', AB is extended to B', and BC is extended to C'

a. What is the sum of angles BAC and ACB and CBA?

b. Angle BAA' is known as the exterior angle to the triangle ABC at A. What is the size of angle BAA' in terms of the angles ABC and BCA?

What is true for the other exterior angles, ACC' and CBB'?

c. If one sums the exterior angles of triangle A, B, and C what number does one get?

Figure 1

2. The diagram below represents a square pyramid whose "lateral faces" are equilateral triangles.

Figure 2

Thus, ABCD is a square and the line segments Ax, Bx, Cx, and Dx all have the same length as AD.

a. Define the "deficit" at a vertex v of a convex polyhedron to be 360 degrees minus the sum of the angles in the faces that meet at v. Determine the deficits at each vertex of the square pyramid above. What is the sum of the deficits added together for all of the vertices?

b. Find the deficits at each vertex of a cube. What is the sum of the deficits added together for all of the vertices of the cube?

c. Determine the sum of the deficits for all of the vertices of a tetrahedron which has all of its edge lengths equal?

d. Make a conjecture based on the examples that studied above?

3. The cube can be used to show that even for such a seeming simple solid there are a variety of angles that one can pay attention to.

a. Given two intersecting planes, the angle formed by these planes is called the dihedral angle between the planes. (Strictly speaking there are several angles. Can you guess which one it is natural to call the dihedral angle?) What is the size of the dihedral angles for a cube?

b. Given a vertex of a cube one can define the solid angle at that vertex. The solid angle of a vertex is the area of the cap cut off from a sphere of radius 1 centered at that vertex cut by the polyhedral cone formed by the edges at that vertex. What is the solid angle at each vertex of a cube?

c. Given a vertex v of a cube one can compute (as we did above) the sum of the angles in the faces that meet at that vertex of the cube. Compute the sum of these angles for the vertices of a cube.

4. Find the dihedral angles, the solid angles, and the sum of the face angles the vertices of the square pyramid in Figure 2.

5. What is the sum of the interior angles of a plane convex quadrilateral? What is the sum of the interior angles of a plane non-self-intersecting quadrilateral? If one adds the sum of the angles between adjacent pairs of sides of a non-plane quadrilateral what is that sum? (As an interesting example try this for two equilateral triangles which are congruent and share an edge, but lie in different planes.)