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

exterior angle

given triangle ABC and A-B-X, then <CBX is an exterior angle of <ABC

remote interior angles

if C-B-Y, then <A and <C are the remote interior angles of <CBX and <ABY

EAT

an exterior angle of a triangle is greater than its 2 remote interior angles

corollary of EAT

if a triangle has a right angle, then its 2 other angles are acute

AAS correspondence

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

AAS theorem

every AAS correspondence is a congruence

HL theorem

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

distance

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

altitude

an altitude of a ∇ is the ⊥ segment joining a vertex of the ∇ with the line containing the opposite side of the vertex