Lines and transversals form a very simple and compact topic. As such, AMC problems dealing solely with lines in synthetic geometry are scarce. Let’s move on to something more relevant to the AMC. Triangles Loads of AMC geometry problems deal with triangles. Because of this, there are a lot of things and properties that you should know about triangles. Let’s deal with the less obscure ones first. Triangle Sum Theorem: Simply enough, the measures of three interior angles of a triangle add up to 180 degrees. Triangle Inequality: Given a triangle with side lengths a, b, and c, with and being any two sides, it must be true that a “a” “b” + b < c, as well as any rearrangement of this inequality. If a + b c, then the triangle becomes a line. ≥ Basic Perimeter/Area of a Triangle: Perimeter = a + b + c, given sides a, b, and c. Area = bh/2, given a base and a height “b” “h.” Pythagorean Theorem: Given a right triangle with legs and with a hypotenuse (longest side) “a” “b” “c,” a2 + b2 = c2. Here’s some more obscure stuff. Angle Bisector Theorem: a triangle with a specific angle bisector, a/b = c/d. Thales’s Theorem: If a right triangle is inscribed in a circle, the hypotenuse is always that circle’s diameter. AMC 12 Only Other Triangle Area Formulas: A = a)(s b)(s c)), given side lengths a, b, and c and that is half the triangle’s perimeter √(s(s – – – “s” A = [bc(sin(A))]/2 given any angle and any adjacent side lengths and “A” “b” “c”
Lines and transversals form a very simple and compact topic. As such, AMC problems dealing solely with lines in synthetic geometry are scarce. Let’s move on to something more relevant to the AMC. Triangles Loads of AMC geometry problems deal with triangles. Because of this, there are a lot of things and properties that you should know about triangles. Let’s deal with the less obscure ones first. Triangle Sum Theorem: Simply enough, the measures of three interior angles of a triangle add up to 180 degrees. Triangle Inequality: Given a triangle with side lengths a, b, and c, with “a” and “b” being any two sides, it must be true that a + b < c, as well as any rearrangement of this inequality. If a + b ≥ c, then the triangle becomes a line. Basic Perimeter/Area of a Triangle: Perimeter = a + b + c, given sides a, b, and c. Area = bh/2, given a base “b” and a height “h.” Pythagorean Theorem: Given a right triangle with legs “a” and “b” with a hypotenuse (longest side) “c,” a2 + b2 = c2. Here’s some more obscure stuff. Angle Bisector Theorem: a triangle with a specific angle bisector, a/b = c/d. Thales’s Theorem: If a right triangle is inscribed in a circle, the hypotenuse is always that circle’s diameter. AMC 12 Only Other Triangle Area Formulas: A = √(s(s – a)(s – b)(s – c)), given side lengths a, b, and c and that “s” is half the triangle’s perimeter A = [bc(sin(A))]/2 given any angle “A” and any adjacent side lengths “b” and “c”
XI-The%20Grand%20Geometry%20Review%20Part%20I.docx