### G.G: Geometry

#### G.G-CO: Congruence

G.G-CO.A: Experiment with transformations in the plane.

G.G-CO.A.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.

G.G-CO.A.2: Represent and describe transformations in the plane 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.

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

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

G.G-CO.A.5: Given a geometric figure and a rotation, reflection, or translation draw the transformed figure. Specify a sequence of transformations that will carry a given figure onto another.

G.G-CO.B: Understand congruence in terms of rigid motions.

G.G-CO.B.6: Use geometric definitions 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.G-CO.B.8: Explain how the criteria for triangle congruence (ASA, AAS, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.

G.G-CO.C: Prove geometric theorems.

G.G-CO.C.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.

G.G-CO.C.10: Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangle 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.

G.G-CO.C.11: Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and rectangles are parallelograms with congruent diagonals.

G.G-CO.D: Make geometric constructions.

G.G-CO.D.12: Make formal geometric constructions with a variety of tools and methods. Constructions include: copying segments; copying angles; bisecting segments; bisecting angles; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.

G.G-CO.D.13: Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle; with a variety of tools and methods.

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

G.G-SRT.A: Understand similarity in terms of similarity transformations.

G.G-SRT.A.1: Verify experimentally the properties of dilations given by a center and a scale factor:

G.G-SRT.A.1a: 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.G-SRT.A.1b: The dilation of a line segment is longer or shorter in the ratio given by the scale factor.

G.G-SRT.A.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.

G.G-SRT.A.3: Use the properties of similarity transformations to establish the AA, SAS, and SSS criterion for two triangles to be similar.

G.G-SRT.B: Prove theorems involving similarity.

G.G-SRT.B.4: Prove theorems about triangles. Theorems include: an interior line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity.

G.G-SRT.B.5: Use congruence and similarity criteria to prove relationships in geometric figures and solve problems utilizing real-world context.

G.G-SRT.C: Define trigonometric ratios and solve problems involving right triangles.

G.G-SRT.C.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.

G.G-SRT.C.8: Use trigonometric ratios (including inverse trigonometric ratios) and the Pythagorean Theorem to find unknown measurements in right triangles utilizing real-world context.

#### G.G-C: Circles

G.G-C.A: Understand and apply theorems about circles.

G.G-C.A.2: Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle.

G.G-C.A.3: Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.

G.G-C.B: Find arc lengths and areas of sectors of circles.

G.G-C.B.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. Convert between degrees and radians.

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

G.G-GPE.A: Translate between the geometric description and the equation for a conic section.

G.G-GPE.A.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.

G.G-GPE.B: Use coordinates to prove geometric theorems algebraically.

G.G-GPE.B.7: Use coordinates to compute perimeters of polygons and areas of triangles and rectangles.

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

G.G-GMD.A: Explain volume formulas and use them to solve problems.

G.G-GMD.A.1: Analyze and verify the formulas for the volume of a cylinder, pyramid, and cone.

G.G-GMD.A.3: Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems utilizing real-world context.

### G.MP: Standards for Mathematical Practices

#### G.MP.1: Make sense of problems and persevere in solving them.

3.1.1: Mathematically proficient students explain to themselves the meaning of a problem, look for entry points to begin work on the problem, and plan and choose a solution pathway. While engaging in productive struggle to solve a problem, they continually ask themselves, “Does this make sense?' to monitor and evaluate their progress and change course if necessary. Once they have a solution, they look back at the problem to determine if the solution is reasonable and accurate. Mathematically proficient students check their solutions to problems using different methods, approaches, or representations. They also compare and understand different representations of problems and different solution pathways, both their own and those of others.

#### G.MP.3: Construct viable arguments and critique the reasoning of others.

3.3.1: Mathematically proficient students construct mathematical arguments (explain the reasoning underlying a strategy, solution, or conjecture) using concrete, pictorial, or symbolic referents. Arguments may also rely on definitions, assumptions, previously established results, properties, or structures. Mathematically proficient students make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. Mathematically proficient students present their arguments in the form of representations, actions on those representations, and explanations in words (oral or written). Students critique others by affirming or questioning the reasoning of others. They can listen to or read the reasoning of others, decide whether it makes sense, ask questions to clarify or improve the reasoning, and validate or build on it. Mathematically proficient students can communicate their arguments, compare them to others, and reconsider their own arguments in response to the critiques of others.

#### G.MP.5: Use appropriate tools strategically.

3.5.1: Mathematically proficient students consider available tools when solving a mathematical problem. They choose tools that are relevant and useful to the problem at hand. Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful; recognizing both the insight to be gained and their limitations. Students deepen their understanding of mathematical concepts when using tools to visualize, explore, compare, communicate, make and test predictions, and understand the thinking of others.

#### G.MP.6: Attend to precision.

3.6.1: Mathematically proficient students clearly communicate to others using appropriate mathematical terminology, and craft explanations that convey their reasoning. When making mathematical arguments about a solution, strategy, or conjecture, they describe mathematical relationships and connect their words clearly to their representations. Mathematically proficient students understand meanings of symbols used in mathematics, calculate accurately and efficiently, label quantities appropriately, and record their work clearly and concisely.

#### G.MP.7: Look for and make use of structure.

3.7.1: Mathematically proficient students use structure and patterns to assist in making connections among mathematical ideas or concepts when making sense of mathematics. Students recognize and apply general mathematical rules to complex situations. They are able to compose and decompose mathematical ideas and notations into familiar relationships. Mathematically proficient students manage their own progress, stepping back for an overview and shifting perspective when needed.

#### G.MP.8: Look for and express regularity in repeated reasoning.

3.8.1: Mathematically proficient students look for and describe regularities as they solve multiple related problems. They formulate conjectures about what they notice and communicate observations with precision. While solving problems, students maintain oversight of the process and continually evaluate the reasonableness of their results. This informs and strengthens their understanding of the structure of mathematics which leads to fluency.

Correlation last revised: 9/15/2020

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