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postulate angle angle similarity (AA~)

if 2 angles of one triangle are ≅ to 2 ∠'s of another triangle then the triangles are similar

theorem side angle side similarity (SAS~)

if one an ∠ of one triangle is ≅ to an angle of another triangle, and the sides including the 2 ∠'s are proportional then the triangles are similar

theorem side side side similarity (SSS~)

if the corresponding sides of 2 triangles are proportional, then the triangles are similar

theorem 7-3

the altitude to the hypotenuse of a right triangle divides the triangle into two triangles that are similar to the original triangle and to each other.

corollary 1 to thm 7-3

the length of the altitude to the hypotenuse of a right triangle is the geometric mean of the lengths of the segments of the hypotenuse

corollary 2 to thm 7-3

the altitude to the hypotenuse of a right triangle seperates the hypotenuse so the length of each leg of the triangle is the geometric mean of the length of the adjacent hypotenuse segment and the length of the hypotenuse

theorem side splitter

if a line is parallel to one side of a triangle and intersects the other 2 sides, then it divides those sides proportionally

corollary to side splitter thm

if 3 parallel lines intersect two transversals, then the segments intercepted on the transversals are proportional

triangle angle bisector thm

if a ray bisects an angle, then it divides the opposite side into 2 segments that are proportional to the other 2 sides of the triangle