 Congruence, Similarity, Right Triangles, and Trigonometry 912.G-CO.1.1 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. 912.G-CO.1.1 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. 912.G-CO.1.2 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).Also assesses:912.G-CO.1.4 Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. Congruence transformations Translations Reflections Rotations Dilations 912.G-CO.1.2 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).Also assesses:912.G-CO.1.4 Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. Congruence transformations Translations Reflections Rotations Dilations 912.G-CO.1.5 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.Also assesses:912.G-CO.1.3 Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself. Transformations Lines of symmetry 912.G-CO.1.5 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.Also assesses:912.G-CO.1.3 Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself. Transformations Lines of symmetry 912.G-CO.2.6 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.Also assesses:912.G-CO.2.7 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.912.G-CO.2.8 Explain how the criteria for triangle congruence (ASA, SAS, SSS, and Hypotenuse-Leg) follow from the definition of congruence in terms of rigid motions. Transformations 912.G-CO.2.6 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.Also assesses:912.G-CO.2.7 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.912.G-CO.2.8 Explain how the criteria for triangle congruence (ASA, SAS, SSS, and Hypotenuse-Leg) follow from the definition of congruence in terms of rigid motions. Transformations 912.G-CO.3.9 Prove theorems about lines and angles; use theorems about lines and angles to solve problems. 912.G-CO.3.9 Prove theorems about lines and angles; use theorems about lines and angles to solve problems. 912.G-CO.3.10 Prove theorems about triangles; use theorems about triangles to solve problems. Isosceles triangles Sides in triangles Angles in triangles Segments in triangles Bisectors in triangles 912.G-CO.3.10 Prove theorems about triangles; use theorems about triangles to solve problems. Isosceles triangles Sides in triangles Angles in triangles Segments in triangles Bisectors in triangles 912.G-CO.3.11 Prove theorems about parallelograms; use theorems about parallelograms to solve problems. 912.G-CO.3.11 Prove theorems about parallelograms; use theorems about parallelograms to solve problems. 912.G-CO.4.12 Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.).Also assesses:912.G-CO.4.13 Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. Lines and angles Triangles Squares Hexagons 912.G-CO.4.12 Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.).Also assesses:912.G-CO.4.13 Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. Lines and angles Triangles Squares Hexagons 912.G-SRT.1.1 Verify experimentally the properties of dilations given by a center and a scale factor. 912.G-SRT.1.1 Verify experimentally the properties of dilations given by a center and a scale factor. 912.G-SRT.1.2 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. Similar figures Triangles Circles 912.G-SRT.1.2 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. Similar figures Triangles Circles 912.G-SRT.1.3 Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.Also assesses:912.G-SRT.2.4 Prove theorems about triangles. Proofs 912.G-SRT.1.3 Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.Also assesses:912.G-SRT.2.4 Prove theorems about triangles. Proofs 912.G-SRT.2.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Congruent triangles Proofs involving congruent triangles Isosceles, equilateral, and right triangles Similar triangles Proofs involving similar triangles 912.G-SRT.2.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Congruent triangles Proofs involving congruent triangles Isosceles, equilateral, and right triangles Similar triangles Proofs involving similar triangles 912.G-SRT.3.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.Also assesses:912.G-SRT.3.6 Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.912.G-SRT.3.7 Explain and use the relationship between the sine and cosine of complementary angles. Pythagorean Theorem Trigonometric ratios Solve a right triangle 912.G-SRT.3.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.Also assesses:912.G-SRT.3.6 Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.912.G-SRT.3.7 Explain and use the relationship between the sine and cosine of complementary angles. Pythagorean Theorem Trigonometric ratios Solve a right triangle
 Circles, Geometric Measurement, and Geometric Properties with Equations 912.G-C.1.1 Prove that all circles are similar. 912.G-C.1.1 Prove that all circles are similar. 912.G-C.1.2 Identify and describe relationships among inscribed angles, radii, and chords. 912.G-C.1.2 Identify and describe relationships among inscribed angles, radii, and chords. 912.G-C.1.3 Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. Constructions Properties of angles 912.G-C.1.3 Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. Constructions Properties of angles 912.G-C.2.5 Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. 912.G-C.2.5 Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. 912.G-GMD.1.1 Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. 912.G-GMD.1.1 Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. 912.G-GMD.1.3 Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. 912.G-GMD.1.3 Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. 912.G-GMD.2.4 Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. 912.G-GMD.2.4 Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. 912.G-GPE.1.1 Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. Complete the square Write equations and graph circles 912.G-GPE.1.1 Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. Complete the square Write equations and graph circles 912.G-GPE.2.4 Use coordinates to prove simple geometric theorems algebraically. 912.G-GPE.2.4 Use coordinates to prove simple geometric theorems algebraically. 912.G-GPE.2.5 Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point). 912.G-GPE.2.5 Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point). 912.G-GPE.2.6 Find the point on a directed line segment between two given points that partitions the segment in a given ratio. 912.G-GPE.2.6 Find the point on a directed line segment between two given points that partitions the segment in a given ratio. 912.G-GPE.2.7 Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. Distance formula Area and perimeter 912.G-GPE.2.7 Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. Distance formula Area and perimeter
 Modeling with Geometry 912.G-MG.1.1 Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder). Pythagorean Theorem 912.G-MG.1.1 Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder). Pythagorean Theorem 912.G-MG.1.2 Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot). 912.G-MG.1.2 Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot). 912.G-MG.1.3 Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios). 912.G-MG.1.3 Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios).
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