Theorem 5-1-1 Perpendicular Bisector Theorem
If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment.
Theorem 5-1-2 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 5-1-3 Angle Bisector Theorem
If a point is on the bisector of an angle, then it is equidistant from the sides of the angle.
Theorem 5-1-4 Converse of the Angle Bisector Theorem
If a point in the interior of an angle is equidistant from the sides of the angle, then it is on the bisector of the angle.
Theorem 5-2-1 Circumcenter Theorem
The circumcenter of a triangle is equidistant from the vertices of the triangle.
Theorem 5-2-2 Incenter Theorem
The incenter of a triangle is equidistant from the sides of the triangle.
Theorem 5-3-1 Centroid Theorem
The centroid of a triangle is located 2/3 of the distance from each vertex to the midpoint of the opposite side.
Theorem 5-4-1 Triangle Midsegment Theorem
A midsegment of a triangle is parallel to a side of a triangle, and its length is half the length of that side.
Theorem 5-5-1
If two sides of a triangle are not congruent, then the larger angle is opposite the longer side.
Theorem 5-5-2
If two angles of a triangle are not congruent, then the longer side is opposite the larger angle.
Theorem 5-5-3 Triangle Inequality Theorem
The sum of any two side lengths of a triange is greater than the third side length.
Theorem 5-6-1 Hinge Theorem
If two sides of one triangle are congruent to two sides of another triangle and the included angles are not congruent, then longer third side is across from the larger included angle.
Theorem 5-6-2 Converse of the Hinge Theorem
If two sides of one triangle are congruent to two sides of another triangle and the third sides are not congruent, then the larger included angle is across from the longer third side.
Theorem 5-7-1 Converse of the Pythagorean Theorem
If the sum of the squares of the lengths of two sides of a triangle is equal to the square of the length of the third side, then the triangle is a right triangle.
Theorem 5-7-2 Pythagorean Inequalities Theorem
In ?ABC, c is the length of the longest side. If c� > a� + b�, then ?ABC is an obtuse triangle. If c� < a� + b�, then ?ABC is an acute triangle.
Theorem 5-8-1 45�-45�-90� Triangle Theorem
In a 45�-45�-90� triangle, both legs are congruent, and the length of the hypotenuse is the length of a leg times ?2.
Theorem 5-8-2 30�-60�-90� Triangle Theorem
In a 30�-60�-90� triangle, the length of the hypotenuse is 2 times the length of the shorter leg, and the length of the longer leg is the length of the shorter leg times ?3.