LAC 28:CXLII.2301. Geometric Reasoning and Logic

A. Experiment with transformations in the plane. 1. Based on the undefined notions of point, line, distance along a line, and distance around a circular arc, know the precise definitions of angle, circle, perpendicular line, parallel line, and line segment. 2. Use a plane. a. Represent transformations with and without technology. b. Describe transformations as functions that take points in the plane as inputs and give other points as outputs. c. Compare transformations that preserve distance and angle to those that do not. 3. Describe the rotations and reflections that map a preimage onto itself when given a rectangle, parallelogram, trapezoid, or regular polygon. 4. Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. 5. Use a geometric figure and a rotation, reflection, translation, or sequence of transformations. a. Draw the transformed figure with and without technology. b. Specify a sequence that will map a given figure onto another. B. Understand congruence in terms of rigid motions. 1. Use geometric descriptions of rigid motions to transform figures. Predict the effect of a given rigid motion on a given figure. Given two figures, use the definition of congruence in terms of rigid motions to determine if they are congruent. 2. Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent. 3. Explain how the criteria for triangle congruence ASA, SAS, and SSS follow from the definition of congruence in terms of rigid motions. C. Prove and apply geometric theorems. 1. Prove and apply theorems about lines and angles. Theorems include but are not limited to vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. 2. Prove and apply theorems about triangles. Theorems include but are not limited to measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. 3. Prove and apply theorems about parallelograms. Theorems include but are not limited to opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and the converse of this theorem; rectangles are parallelograms with congruent diagonals and the converse of this theorem. D. Make geometric constructions. 1. Make formal geometric constructions with a variety of tools and methods, with or without technology, of an equilateral triangle, a square, and a regular hexagon inscribed in a circle.