Congruent Triangles through Transformations

Transformations and Congruence

• Rotations, reflections, and translations of a geometric figure preserve congruence, similar to how properties of operations preserve equivalence of arithmetic and algebraic expressions.
• A sequence of transformations can result in a congruent triangle, and this can be demonstrated using physical models, transparencies, geometric software, or other tools.

Properties of Transformations

• Rotations, reflections, and translations preserve congruence because they preserve:
  • Line segments of the same length
  • Angles of the same measure
  • Parallel lines
• These properties can be verified experimentally.

Congruence and Similarity

• A two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations.
• A two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations.

Using Coordinates

• The effect of dilations, translations, rotations, and reflections on two-dimensional figures can be described using coordinates.

Angle Properties

• The angle sum and exterior angle of triangles can be established using informal arguments.
• The angles created when parallel lines are cut by a transversal can be understood using transversals.
• The angle-angle criterion for similarity of triangles can be demonstrated using arrangements of triangles.

Explore how geometric transformations preserve congruence in triangles, using critical thinking and problem-solving skills. Use physical models and software to demonstrate a sequence of transformations resulting in a congruent triangle.