{
 "source": "https://peabody.kids/flashcards/math-geometry-theorems",
 "id": "math-geometry-theorems",
 "title": "Geometry Theorems and Rules",
 "category": "math",
 "subject": "Math",
 "levels": [
  "9-12"
 ],
 "description": "Triangle, circle and polygon theorems and congruence rules.",
 "tags": [
  "theorems",
  "geometry",
  "proof"
 ],
 "frontLabel": "Name",
 "backLabel": "Statement",
 "cards": [
  [
   "Triangle angle sum theorem",
   "The three angles of a triangle add up to 180°"
  ],
  [
   "Exterior angle theorem",
   "An exterior angle of a triangle equals the sum of the two remote interior angles"
  ],
  [
   "Isosceles triangle theorem",
   "Angles opposite equal sides are equal"
  ],
  [
   "Converse of the isosceles triangle theorem",
   "Sides opposite equal angles are equal"
  ],
  [
   "Triangle inequality theorem",
   "The sum of any two sides is greater than the third side"
  ],
  [
   "Pythagorean theorem",
   "In a right triangle, a² + b² = c²"
  ],
  [
   "Converse of the Pythagorean theorem",
   "If a² + b² = c², the triangle is a right triangle"
  ],
  [
   "45-45-90 triangle",
   "Sides are in the ratio 1 : 1 : √2"
  ],
  [
   "30-60-90 triangle",
   "Sides are in the ratio 1 : √3 : 2"
  ],
  [
   "Midsegment theorem",
   "A segment joining midpoints of two sides is parallel to the third side and half as long"
  ],
  [
   "Triangle angle bisector theorem",
   "An angle bisector splits the opposite side in the ratio of the adjacent sides"
  ],
  [
   "SSS congruence",
   "Three pairs of equal sides make triangles congruent"
  ],
  [
   "SAS congruence",
   "Two sides and the angle between them equal makes triangles congruent"
  ],
  [
   "ASA congruence",
   "Two angles and the side between them equal makes triangles congruent"
  ],
  [
   "AAS congruence",
   "Two angles and a non-included side equal makes triangles congruent"
  ],
  [
   "HL congruence",
   "Right triangles with equal hypotenuse and one equal leg are congruent"
  ],
  [
   "CPCTC",
   "Corresponding parts of congruent triangles are congruent"
  ],
  [
   "AA similarity",
   "Two triangles with two equal angles are similar"
  ],
  [
   "SAS similarity",
   "Two proportional sides with equal included angle make triangles similar"
  ],
  [
   "SSS similarity",
   "Three proportional sides make triangles similar"
  ],
  [
   "Vertical angles theorem",
   "Vertical angles are equal"
  ],
  [
   "Linear pair postulate",
   "Two angles that form a straight line add up to 180°"
  ],
  [
   "Corresponding angles postulate",
   "Parallel lines cut by a transversal make equal corresponding angles"
  ],
  [
   "Alternate interior angles theorem",
   "Parallel lines cut by a transversal make equal alternate interior angles"
  ],
  [
   "Same-side interior angles theorem",
   "Parallel lines cut by a transversal make supplementary same-side interior angles"
  ],
  [
   "Perpendicular bisector theorem",
   "A point on the perpendicular bisector is equally far from the segment's endpoints"
  ],
  [
   "Angle bisector theorem (points)",
   "A point on an angle bisector is equally far from the angle's sides"
  ],
  [
   "Centroid",
   "The point where a triangle's three medians meet"
  ],
  [
   "Circumcenter",
   "The point where perpendicular bisectors meet; the center of the circumscribed circle"
  ],
  [
   "Incenter",
   "The point where angle bisectors meet; the center of the inscribed circle"
  ],
  [
   "Orthocenter",
   "The point where a triangle's altitudes meet"
  ],
  [
   "Centroid ratio",
   "The centroid divides each median in a 2 : 1 ratio, longer part toward the vertex"
  ],
  [
   "Inscribed angle theorem",
   "An inscribed angle is half its intercepted arc"
  ],
  [
   "Central angle theorem",
   "A central angle equals its intercepted arc"
  ],
  [
   "Thales' theorem",
   "An angle inscribed in a semicircle is a right angle"
  ],
  [
   "Tangent-radius theorem",
   "A tangent is perpendicular to the radius at the point of tangency"
  ],
  [
   "Two tangents theorem",
   "Tangent segments from one outside point to a circle are equal"
  ],
  [
   "Chord-chord theorem",
   "If two chords cross, the products of their pieces are equal"
  ],
  [
   "Opposite angles in a cyclic quadrilateral",
   "They add up to 180°"
  ],
  [
   "Parallelogram theorem: sides",
   "Opposite sides are equal"
  ],
  [
   "Parallelogram theorem: angles",
   "Opposite angles are equal; consecutive angles add up to 180°"
  ],
  [
   "Parallelogram theorem: diagonals",
   "Diagonals bisect each other"
  ],
  [
   "Rectangle diagonals",
   "Diagonals of a rectangle are equal"
  ],
  [
   "Rhombus diagonals",
   "Diagonals are perpendicular and bisect the angles"
  ],
  [
   "Kite diagonals",
   "Diagonals are perpendicular"
  ],
  [
   "Isosceles trapezoid base angles",
   "Base angles of an isosceles trapezoid are equal"
  ],
  [
   "Side-splitter theorem",
   "A line parallel to one side of a triangle divides the other two sides proportionally"
  ],
  [
   "Geometric mean (altitude) theorem",
   "The altitude to the hypotenuse of a right triangle is the geometric mean of the two segments it makes"
  ],
  [
   "Law of sines",
   "a/sin A = b/sin B = c/sin C"
  ],
  [
   "Law of cosines",
   "c² = a² + b² − 2ab cos C"
  ],
  [
   "Euler line",
   "The centroid, circumcenter and orthocenter of a triangle lie on one line"
  ],
  [
   "Ceva's theorem",
   "Cevians through a point in a triangle satisfy (AF/FB)(BD/DC)(CE/EA) = 1"
  ],
  [
   "Pick's theorem",
   "Area of a lattice polygon = i + b/2 − 1, with i inside points and b boundary points"
  ],
  [
   "Euler's polyhedron formula",
   "V − E + F = 2"
  ]
 ]
}