Theorems 5-1 to 5-12 and Corollary
Theorem 5-5: Converse of the Angle Bisector Theorem
If a point in the interior of an angle is equidistant from the sides od the angle, then the point is on the angle bisector.
Theorem 5-3: Converse of the Perpendicular Bisector Theorem
If a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the segment.
Theorem 5-4: Angle Bisector Theorem
If a point is on the bisector of an angle, then the point is equidistant from the sides of the angle.
Theorem 5-2: Perpendicular Bisector Theorem
If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segments.
Theorem 5-1: Triangle Midsegment Theorem
If a segment joins the midpoint of two sides of a triangle, then the segment is parallel to the third side and is half as long.
Theorem 5-10
If two sides of a triangle are mot congruent, then the larger angles lies opposite the longer side.
Theorem 5-11
If two sides of a triangle are not congruent, then the longer side lies on the longer angle.
Theorem 5-7: Concurrency of Angle Bisectors Theorem
The bisectors of the angles of a triangle are concurrent at a point equidistant from the sides of the triangle.
Theorem 5-9: Concurrency of Altitudes Theorem
The lines that contain the altitudes of a triangle are concurrent.
Corollary to the Triangle Exterior Angle Theorem
The measure of an exterior angle of a triangle is greater than the measure of each of its remote interior angles.
Theorem 5-8: Concurrency of Medians Theorem
The medians of a triangle are concurrent at a point (the centroid of the triangle) that is two thirds the distance from each vertex to the midpoint of the opposite side.
Theorem 5-6: Concurrency of Perpendicular Bisector Theorem
The perpendicular bisector of the sides of a triangle are concurrent at a point equidistant from the vertices.
Theorem 5-12: Triangle Inequality Theorem
The sum of the lengths of any two sides of a triangle is greater than the length of the third side.