### G: Geometry

#### G-CO: Congruence

1.1.1: Experiment with transformations in the plane.

G-CO.1: Students will: 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.

G-CO.2: Students will: 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).

G-CO.3: Students will: Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.

G-CO.4: Students will: Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.

G-CO.5: Students will: 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.

1.1.2: Understand congruence in terms of rigid motions.

G-CO.6: Students will: 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.

G-CO.7: Students will: 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.

G-CO.8: Students will: Explain how the criteria for triangle congruence, angle-side-angle (ASA), side-angle-side (SAS), and side-side-side (SSS), follow from the definition of congruence in terms of rigid motions.

1.1.3: Prove geometric theorems.

G-CO.9: Students will: Prove theorems about lines and angles.

G-CO.10: Students will: Prove theorems about triangles.

G-CO.11: Students will: Prove theorems about parallelograms.

1.1.4: Make geometric constructions.

G-CO.12: Students will: Make formal geometric constructions with a variety of tools and methods such as compass and straightedge, string, reflective devices, paper folding, and dynamic geometric software.

G-CO.13: Students will: Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.

#### G-SRT: Similarity, Right Triangles, and Trigonometry

1.2.1: Understand similarity in terms of similarity transformations.

G-SRT.14: Students will: Verify experimentally the properties of dilations given by a center and a scale factor.

G-SRT.14.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.

G-SRT.14.b: The dilation of a line segment is longer or shorter in the ratio given by the scale factor.

G-SRT.15: Students will: 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.

G-SRT.16: Students will: Use the properties of similarity transformations to establish the angle-angle (AA) criterion for two triangles to be similar.

1.2.2: Prove theorems involving similarity.

G-SRT.17: Students will: Prove theorems about triangles.

G-SRT.18: Students will: Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.

1.2.3: Define trigonometric ratios and solve problems involving right triangles.

G-SRT.19: Students will: 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.

G-SRT.20: Students will: Explain and use the relationship between the sine and cosine of complementary angles.

G-SRT.21: Students will: Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.

#### G-C: Circles

1.3.1: Understand and apply theorems about circles.

G-C.24: Students will: Prove that all circles are similar.

G-C.25: Students will: Identify and describe relationships among inscribed angles, radii, and chords.

G-C.26: Students will: Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.

1.3.2: Find arc lengths and areas of sectors of circles.

G-C.28: Students will: 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.

#### G-GPE: Expressing Geometric Properties with Equations

1.4.1: Translate between the geometric description and the equation for a conic section.

G-GPE.29: Students will: 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.

1.4.2: Use coordinates to prove simple geometric theorems algebraically.

G-GPE.31: Students will: 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).

G-GPE.33: Students will: Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.

#### G-GMD: Geometric Measurement and Dimension

1.5.1: Explain volume formulas and use them to solve problems.

G-GMD.35: Students will: Give an informal argument for the formulas for the circumference of a circle; area of a circle; and volume of a cylinder, pyramid, and cone.

G-GMD.36: Students will: Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.

1.5.2: Visualize relationships between two-dimensional and three-dimensional objects.

G-GMD.38: Students will: Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects.

#### G-MG: Modeling with Geometry

1.6.1: Apply geometric concepts in modeling situations.

G-MG.39: Students will: Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).

G-MG.40: Students will: Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, British Thermal Units (BTUs) per cubic foot).

G-MG.41: Students will: 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).

### S: Statistics and Probability

#### S-MD: Using Probability to Make Decisions

2.1.1: Use probability to evaluate outcomes of decisions.

S-MD.42: Students will: Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator).

S-MD.43: Students will: Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game).

Correlation last revised: 3/17/2020

This correlation lists the recommended Gizmos for this state's curriculum standards. Click any Gizmo title below for more information.