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Understand congruence and similarity using physical models, pictorial representations, transparencies, or geometry software. [P] [L] [R]

1.   Verify experimentally the properties of rotations, reflections, and translations:

a.   Lines are taken to lines, and line segments to line segments of the same length.

b.   Angles are taken to angles of the same measure. [P]

c.   Parallel lines are taken to parallel lines.[P]

2.   Understand that 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, describe a sequence that exhibits the congruence between them. [L]

3.   Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates. [A]

4.   Understand that 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, describe a sequence that exhibits the similarity between them. [R]

5.   Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles. For example, arrange three copies of the same triangle so thatthe sum of the three angles appears to form a line, and give an argumentin terms of transversals why this is so.

 Congruence                                                                                                                   G-CO Experiment with transformations in the plane 1.   Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. 2.   Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch). 3.   Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself. 4.   Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. 5.   Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another. Understand congruence in terms of rigid motions 6.   Use geometric descriptions of rigid motions to transform figures and to 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 decide if they are congruent. 7.   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. 8.   Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions. Prove geometric theorems 9.   Prove theorems about lines and angles. Theorems include: 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. 10. Prove theorems about triangles. Theorems include: 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. Know and use the triangle inequality theorem. 11. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Make geometric constructions 12. Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment;copying an angle; bisecting a segment; bisecting an angle; constructingperpendicular lines, including the perpendicular bisector of a line segment;and constructing a line parallel to a given line through a point not on theline. 13. Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.
 Similarity, Right Triangles, and Trigonometry                                                              G-SRT Understand similarity in terms of similarity transformations 1.   Verify experimentally the properties of dilations given by a center and a scale factor: a.   A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. b.   The dilation of a line segment is longer or shorter in the ratio givenby the scale factor. 2.   Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. 3.   Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. Prove theorems involving similarity 4.   Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. 5.   Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.