Properties of Rotations, Reflections, and Translations

Properties of Transformations

• Rotations, reflections, and translations take lines to lines, and line segments to line segments of the same length.
• Angles are taken to angles of the same measure.
• Parallel lines are taken to parallel lines.

Congruence of Figures

• 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.
• Given two congruent figures, a sequence can be described to exhibit the congruence between them.

Effect of Transformations on Figures

• Dilations, translations, rotations, and reflections can be described using coordinates to show their effect on two-dimensional figures.

Similarity of Figures

• 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.
• Given two similar two-dimensional figures, a sequence can be described to exhibit the similarity between them.

Angle Properties

• Informal arguments can be used to establish facts about the angle sum and exterior angle of triangles.
• The angle-angle criterion for similarity of triangles can be demonstrated.

Critical Thinking and Problem Solving

• 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 be explained to result in a congruent or similar triangle.

Mathematical Practices

• Physical models, transparencies, geometric software, or other tools can be used to explore the relationships between transformations and congruence and similarity.
• The structure of the coordinate system can be used to describe the locations of figures obtained with rotations, reflections, and translations.
• It can be reasoned that since any one rotation, reflection, or translation of a figure preserves congruence, then any sequence of those transformations must also preserve congruence.

Verify and demonstrate properties of geometric transformations, including congruent figures and preservation of lines, line segments, angles, and parallel lines.