Congruent Triangles through Transformations

Study Notes

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.

Description

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.