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Geometry Theorems and Rules

Triangle, circle and polygon theorems and congruence rules.

54 cardsGrades 9–12#theorems#geometry#proof

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54 cards
  • Triangle angle sum theoremName
    The three angles of a triangle add up to 180°
  • Exterior angle theoremName
    An exterior angle of a triangle equals the sum of the two remote interior angles
  • Isosceles triangle theoremName
    Angles opposite equal sides are equal
  • Converse of the isosceles triangle theoremName
    Sides opposite equal angles are equal
  • Triangle inequality theoremName
    The sum of any two sides is greater than the third side
  • Pythagorean theoremName
    In a right triangle, a² + b² = c²
  • Converse of the Pythagorean theoremName
    If a² + b² = c², the triangle is a right triangle
  • 45-45-90 triangleName
    Sides are in the ratio 1 : 1 : √2
  • 30-60-90 triangleName
    Sides are in the ratio 1 : √3 : 2
  • Midsegment theoremName
    A segment joining midpoints of two sides is parallel to the third side and half as long
  • Triangle angle bisector theoremName
    An angle bisector splits the opposite side in the ratio of the adjacent sides
  • SSS congruenceName
    Three pairs of equal sides make triangles congruent
  • SAS congruenceName
    Two sides and the angle between them equal makes triangles congruent
  • ASA congruenceName
    Two angles and the side between them equal makes triangles congruent
  • AAS congruenceName
    Two angles and a non-included side equal makes triangles congruent
  • HL congruenceName
    Right triangles with equal hypotenuse and one equal leg are congruent
  • CPCTCName
    Corresponding parts of congruent triangles are congruent
  • AA similarityName
    Two triangles with two equal angles are similar
  • SAS similarityName
    Two proportional sides with equal included angle make triangles similar
  • SSS similarityName
    Three proportional sides make triangles similar
  • Vertical angles theoremName
    Vertical angles are equal
  • Linear pair postulateName
    Two angles that form a straight line add up to 180°
  • Corresponding angles postulateName
    Parallel lines cut by a transversal make equal corresponding angles
  • Alternate interior angles theoremName
    Parallel lines cut by a transversal make equal alternate interior angles
  • Same-side interior angles theoremName
    Parallel lines cut by a transversal make supplementary same-side interior angles
  • Perpendicular bisector theoremName
    A point on the perpendicular bisector is equally far from the segment's endpoints
  • Angle bisector theorem (points)Name
    A point on an angle bisector is equally far from the angle's sides
  • CentroidName
    The point where a triangle's three medians meet
  • CircumcenterName
    The point where perpendicular bisectors meet; the center of the circumscribed circle
  • IncenterName
    The point where angle bisectors meet; the center of the inscribed circle
  • OrthocenterName
    The point where a triangle's altitudes meet
  • Centroid ratioName
    The centroid divides each median in a 2 : 1 ratio, longer part toward the vertex
  • Inscribed angle theoremName
    An inscribed angle is half its intercepted arc
  • Central angle theoremName
    A central angle equals its intercepted arc
  • Thales' theoremName
    An angle inscribed in a semicircle is a right angle
  • Tangent-radius theoremName
    A tangent is perpendicular to the radius at the point of tangency
  • Two tangents theoremName
    Tangent segments from one outside point to a circle are equal
  • Chord-chord theoremName
    If two chords cross, the products of their pieces are equal
  • Opposite angles in a cyclic quadrilateralName
    They add up to 180°
  • Parallelogram theorem: sidesName
    Opposite sides are equal
  • Parallelogram theorem: anglesName
    Opposite angles are equal; consecutive angles add up to 180°
  • Parallelogram theorem: diagonalsName
    Diagonals bisect each other
  • Rectangle diagonalsName
    Diagonals of a rectangle are equal
  • Rhombus diagonalsName
    Diagonals are perpendicular and bisect the angles
  • Kite diagonalsName
    Diagonals are perpendicular
  • Isosceles trapezoid base anglesName
    Base angles of an isosceles trapezoid are equal
  • Side-splitter theoremName
    A line parallel to one side of a triangle divides the other two sides proportionally
  • Geometric mean (altitude) theoremName
    The altitude to the hypotenuse of a right triangle is the geometric mean of the two segments it makes
  • Law of sinesName
    a/sin A = b/sin B = c/sin C
  • Law of cosinesName
    c² = a² + b² − 2ab cos C
  • Euler lineName
    The centroid, circumcenter and orthocenter of a triangle lie on one line
  • Ceva's theoremName
    Cevians through a point in a triangle satisfy (AF/FB)(BD/DC)(CE/EA) = 1
  • Pick's theoremName
    Area of a lattice polygon = i + b/2 − 1, with i inside points and b boundary points
  • Euler's polyhedron formulaName
    V − E + F = 2

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