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the parts theorem
1. if D E segment AB, then segment DA < segment AB and segment DB < segment AB; 2. if D E int(<AOB), then <AOD < <AOB and <DOB < <AOB
given triangle ABC and triangle XYZ with <A congruent to <X, <B congruent to <Y, and BC = YZ then the correspondence between the 2 triangles is an AAS correspondence
if the hypotenuse and one leg of a right triangle are congruent to the hypotenuse and one leg of another right triangle, then the triangles are congruent
</side theorem 1
if 2 sides of a triangle are not congruent, then the angles opposite them are not congruent, and the larger angle is opposite the longer side
</side theorem 2
if 2 angles of a triangle are not congruent, then the sides opposite them are not congruent, and the longer side is opposite the larger <
the first minimum theorem
the shortest segment joining a point to a line is the perpendicular segment
the distance between a line and a point not on the line is the length of the perpendicular segment joining the point and the line. the distance between a line and a point on the line is zero
triangle inequality theorem
the sum of the lengths of any 2 sides of a triangle is greater than the length of the third side
the hinge theorem
if 2 sides of ∇1 are ≅ to 2 sides of ∇2, and the included ∠ of ∇1 is greater than the included ∠ of ∇2, then the third side of ∇1 is longer than the third side of ∇2
the converse hinge theorem
if 2 sides of ∇1 are ≅ to 2 sides of ∇2, and the third side of ∇1 is greater than the thrid side of ∇2, then the ∠ opposite the third side of ∇1 is larger than the ∠ opposite the third side of ∇2
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