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.

**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
by^{T3}

**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*.

6. Parallel Lines and Similar and Congruent Triangles