Congruence Experiment with Transformations in the Plane GCO.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.
GCO.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).
GCO.3. Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.
GCO.4. Develop definitions of rotations, reflections and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.
GCO.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 GCO.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.
GCO.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.
GCO.8. Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.
Understand Congruence in Terms of Rigid Motions GCO.9. Prove theorems about lines and angles.
GCO.10. Prove theorems about triangles
GCO.11. Prove theorems about parallelograms.
Make Geometric Constructions GCO.12. Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.).
GCO.13. Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.
Similarity, Right Triangles, and Trigonometry Understand Similarity in Terms of Similarity Transformations GSRT.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 given by the scale factor.
GSRT.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.
GSRT.3. Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.
Prove Theorems Involving Similarity GSRT.4. Prove theorems about triangles.
GSRT.5. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.
Define Trigonometric Ratios and Solve Problems Involving Right Triangles GSRT.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.
GSRT.7. Explain and use the relationship between the sine and cosine of complementary angles.
GSRT.8. Use Trigonometric ratios and the Pythagorean theorem to solve right triangles in applied problems. Apple Trigonometry to General Triangles GSRT.9. Derive the formula A – ½ ab sin (C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side.
GSRT.10. Prove the Law of Sines and Law of Cosines and use them to solve problems.
GSRT.11. Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and nonright triangles (e.g. surveying problems, resultant forces).
Circles Understand and Apply Theorems about Circles GC.1. Prove that all circles are similar.
GC.2. Identify and describe relationships among inscribed angles, radii, and chords.
GC.3. Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.
GC.4. Construct a tangent line from a point outside a given circle to the circle.
Find Arc Lengths and Areas of Sectors of Circles GC.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.
Expressing Geometric Properties with Equations Translate Between the Geometric Description and the Equation for a Conic Section GGPE.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.
GGPE.2. Derive the equation of a parabola given a focus and directrix.
GGPE.3. Derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or difference of distances fro the foci is constant.
Use Coordinates to Prove Simple Geometric Theorems Algebraically GGPE.4. Use coordinates to prove simple geometric theorems algebraically.
GGPE.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).
GGPE.6. Find the point on a directed line segment between two given points that partitions the segment in a given ratio.
GGPE.7. Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g. using the distance formula.
Geometric Measurement and Dimension Explain Volume Formulas and Use Them to Solve Problems GGMD.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.
GGMD.2. Give an informal argument using Cavalieri’s principle for the formulas for the volume of a sphere and other solid figures.
GGMD.3. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.
Visualize Relationships Between TwoDimensional and ThreeDimensional Objects IA.7. Plot points in threedimensions
GGMD.4. Identify the shapes of twodimensional crosssections of threedimensional objects, and identify threedimensional objects.
Modeling with Geometry Apply Geometric Concepts in Modeling Situations GMG.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.
GMG.2. Apply concepts of density based on area and volume in modeling situations e.g. person per square mile, BTUs per cubic foot.
GMG.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. Use Diagrams Consisting of Vertices and Edges (VertexEdge Graphs) to Model and Solve Problems Related to Networks (IA) IA.8. Understand, analyze, evaluate, and apply vertexedge graphs to model and solve problems related to paths, circuits, networks, and relationships among a finite number of elements, in realworld and abstract settings.
IA.9. Model and solve problems using at least two of the following fundamental graph topics and models: Euler paths and circuits, Hamilton path and circuits, the traveling salesman problem (TSP), minimum spanning trees, critical paths, vertex coloring.
IA.10. Compare and contrast vertexedge graph topics and models in terms of: a. Properties b. Algorithms c. Optimization d. Types of problems that can be solved

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