Geometry, Plane Geometry, Triangles, Triangle Congruence

Triangle Congruence deals with determining if two triangles have the exact same size and shape. Questions involve applying congruence postulates (like SSS, SAS, ASA, AAS) to prove triangles are congruent and to deduce equality of corresponding sides and angles.

• Question

  Cut a square into three parts, and then assemble them into an acute triangle with all sides of different lengths.

  Topics:

  Geometry -> Plane Geometry -> Triangles -> Triangle Congruence Combinatorics -> Combinatorial Geometry -> Cut a Shape / Dissection Problems

• Question

  Two congruent triangles form a Star of David as depicted in the drawing. Prove that the shaded area is equal to the hatched area.

  

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  Topics:

  Geometry -> Area Calculation Proof and Example Set Theory Geometry -> Plane Geometry -> Triangles -> Triangle Congruence

• Pentagon

  In a convex pentagon `ABCDE`, the following holds: `AE=AD`, `AB=AC`, and `angle CAD=angle ABE + angle AEB`.

  Let `AM` be the median to the side `BE` in the triangle `ABE`. Prove that `AM` is half the length of the segment `CD`.

  Topics:

  Geometry -> Plane Geometry -> Angle Calculation Geometry -> Plane Geometry -> Triangles -> Triangle Congruence Geometry -> Plane Geometry -> Plane Transformations -> Congruence Transformations (Isometries) -> Rotation
  Sources:
  • Tournament of Towns, 1983-1984, Fall, Practice Version, Grades 9-10 Question 2 Points 3

• Question

  Given a regular polygon with n vertices. Calculate the number of distinct (non-congruent) triangles whose vertices coincide with the vertices of the polygon.

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  Topics:

  Combinatorics Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules Geometry -> Plane Geometry -> Angle Calculation Geometry -> Plane Geometry -> Triangles -> Triangle Congruence
  Sources:
  • Mink Exercises and Additional Competition Materials, 2018-2019, Exercise 4 Question 3

• Where is the point?

  In a convex hexagon ABCDEF, triangles ACE and BDF are congruent and regular. Show that the three segments connecting the midpoints of opposite sides of the hexagon intersect at one point.

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  Topics:

  Geometry -> Plane Geometry -> Symmetry Geometry -> Plane Geometry -> Triangles -> Triangle Congruence Geometry -> Vectors
  Sources:
  • Gillis Mathematical Olympiad, 2019-2020 Question 3

• Triangles on Lines

  Six congruent isosceles triangles are arranged as shown in the figure.
  Show that points C, F, and M are collinear.

  

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  Topics:

  Geometry -> Plane Geometry -> Angle Calculation Geometry -> Plane Geometry -> Triangles -> Triangle Congruence Geometry -> Plane Geometry -> Plane Transformations -> Congruence Transformations (Isometries) -> Parallel Translation / Translation
  Sources:
  • Gillis Mathematical Olympiad, 2018-2019 Question 3