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Secondary Mathematics I
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Standard G.CO.8
Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.
  • Congruence
    The Utah State Office of Education (USOE) and educators around the state of Utah developed these guides for the Secondary Mathematics 1 Cluster "Experiment with transformations in the plane" / Standards G.CO.1, G.CO.2, G.CO.3, G.CO.4 and G.CO.5 and Cluster "Understand congruence in terms of rigid motions" / Standards G.CO.6, G.CO.7 and G.CO.8 and Cluster "Make geometric constructions" / Standards G.CO.12 and G.CO.13.
  • Properties of Congruent Triangles
    The goal of this task is to understand how congruence of triangles, defined in terms of rigid motions, relates to the corresponding sides and angles of these triangles.
  • Reflections and Equilateral Triangles
    This activity is one in a series of tasks using rigid transformations of the plane to explore symmetries of classes of triangles, with this task in particular focusing on the class of equilaterial triangles.
  • Reflections and Equilateral Triangles II
    This task examines some of the properties of reflections of the plane which preserve an equilateral triangle: these were introduced in ''Reflections and Isosceles Triangles'' and ''Reflection and Equilateral Triangles I''.
  • Reflections and Isosceles Triangles
    This activity is one in a series of tasks using rigid transformations of the plane to explore symmetries of classes of triangles, with this task in particular focussing on the class of isosceles triangles.
  • When Does SSA Work to Determine Triangle Congruence?
    The triangle congruence criteria, SSS, SAS, ASA, all require three pieces of information. It is interesting, however, that not all three pieces of information about sides and angles are sufficient to determine a triangle up to congruence. In this problem, we considered SSA. Also insufficient is AAA, which determines a triangle up to similarity. Unlike SSA, AAS is sufficient because two pairs of congruent angles force the third pair of angles to also be congruent.
  • Why Does ASA Work?
    The two triangles in this problem share a side so that only one rigid transformation is required to exhibit the congruence between them. In general more transformations are required and the "Why does SSS work?'' and "Why does SAS work?'' problems show how this works.
  • Why does SAS work?
    For these particular triangles, three reflections were necessary to express how to move from ABC to DEF. Sometimes, however, one reflection or two reflections will suffice. Since any rigid motion will take triangle ABC to a congruent triangle DEF, this shows the remarkable fact that any rigid motion of the plane can be expressed as one reflection, a composition of two reflections, or a composition of three reflections.
  • Why does SSS work?
    This particular sequence of transformations which exhibits a congruency between triangles ABC and DEF used one translation, one rotation, and one reflection.
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