### 5. Arc Length, Angles, and Rotations

Theorems 5.1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13

Theorem 5.1: If two distinct points on a circle are not diametrically opposed, then the tangent lines at those points meet a point outside the circle.

Theorem 5.2: Let   P   be a partition of the arc between   A   and   B.   Then the inside approximation corresponding to   P   is shorter than the outside approximation. If   Q   is a refinement of   P.   Then the inside approximation corresponding to   Q   is larger than the inside approximation corresponding to   P,   and the outside approximation corresponding to   Q   is less than the outside approximation corresponding to   P.

Theorem 5.3: Arc length is preserved by isometries.1

Theorem 5.4: (The angle addition axiom) Let   C   be a point on the arc between   A   and   B.   Then the length of the arc from   A   to   B   is the sum of the lengths of the arcs from   A   to   C   and from   C   to   B.2

Theorem 5.5: If a circle is drawn at the vertex of an angle, then the arc length between the points on the circle where the arms of the angle meet the circle is proportional to the radius of the circle.

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Theorem 5.6: The number of radians in an angle is preserved by isometries.

Theorem 5.7: The number of radians in a straight angle is half of the number of radians in a full circle.

Theorem 5.8: The number of radians in a right angle is half of the number of radians in a straight angle.

Theorem 5.9: The length of the arc of the unit circle above the points   a   and   c   on the   x-axis is given byT3

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Theorem 5.10: The number of radians in an angle is a continuous function of the   x-coordinate of a point on the moveable arm of the angle kept at a fixed distance from the vertex.

Theorem 5.11: Vertical angles are the same size.

Theorem 5.12: A rotation is a composition of two reflections, and hence is an invertible isometry.3

Theorem 5.13: (The Protractor Axiom) Let    x   be a positive real number less than   pi. Let   A   and   B   be two points. Then there are exactly two lines going through point   A   which make an angle of   x   radians with the ray from   A   through   B.   In one, the angle would be measured clockwise from   A,   and in the other, the angle would be measured counterclockwise from   A.   The two angles are reflections of each other about the line determined by   AB.

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6. Parallel Lines and Similar and Congruent Triangles