Shape

From only a few axioms, the deductive method of Euclid generates a rich body of theorems about geometric objects, their attributes and relationships. Once understood, those attributes and relationships can be applied in diverse practical situations—interpreting a schematic drawing, estimating the amount of wood needed to frame a sloping roof, rendering computer graphics, or designing a sewing pattern for the most efficient use of material. Understanding the attributes of geometric objects often relies on measurement: a circle is a set of points in a plane at a fixed distance from a point; a cube is bounded by six squares of equal area; when two parallel lines are crossed by a transversal, pairs of corresponding angles are congruent. The concepts of congruence, similarity and symmetry can be united under the concept of geometric transformation. Reflections and rotations each explain a particular type of symmetry, and the symmetries of an object offer insight into its attributes—as when the reflective symmetry of an isosceles triangle assures that its base angles are congruent. Applying a scale transformation to a geometric figure yields a similar figure. The transformation preserves angle measure, and lengths are related by a constant of proportionality. If the constant of proportionality is one, distances are also preserved (so the transformation is a rigid transformation) and the figures are congruent. The definitions of sine, cosine and tangent for acute angles are founded on right triangle similarity, and, with the Pythagorean theorem, are fundamental in many practical and theoretical situations. //Connections to Coordinates, Functions and Modeling.// The Pythagorean theorem is a key link between geometry, measurement and distance in the coordinate plane. Parameter changes in families of functions can be interpreted as transformations applied to their graphs and those functions, as well as geometric objects in their own right, can be used to model contextual situations.

Students understand that:
 * Core Concepts**
 * 1) Shapes and their parts, attributes, and measurements can be analyzed deductively.
 * 2) Congruence, similarity, and symmetry can be analyzed using transformations.
 * 3) Mathematical shapes model the physical world, resulting in practical applications of geometry.
 * 4) Right triangles and the Pythagorean theorem are central to geometry and its applications, including trigonometry.

Students can and do:
 * Core Skills**
 * 1) Use multiple geometric properties to solve problems involving geometric figures.
 * 2) Prove theorems, test conjectures and identify logical errors.
 * 3) Construct and interpret representations of geometric objects.
 * 4) Solve problems involving measurements.
 * 5) Solve problems involving similar triangles and scale drawings.
 * 6) Apply properties of right triangles and right triangle trigonometry to solve problems.