### G.CO: Congruence

#### 1.1: Experiment with transformations in the plane.

G.CO.1: Verify experimentally (for example, using patty paper or geometry software) the properties of rotations, reflections, translations, and symmetry:

G.CO.1a: Lines are taken to lines, and line segments to line segments of the same length.

G.CO.1b: Angles are taken to angles of the same measure.

G.CO.1c: Parallel lines are taken to parallel lines.

G.CO.2: Recognize transformations as functions that take points in the plane as inputs and give other points as outputs and describe the effect of translations, rotations, and reflections on two-dimensional figures.

#### 1.2: Understand congruence in terms of rigid motions.

G.CO.3: Given two congruent figures, describe a sequence of rigid motions that exhibits the congruence (isometry) between them using coordinates and the non-coordinate plane.

G.CO.5: Given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent.

G.CO.6: Demonstrate triangle congruence using rigid motion (ASA, SAS, and SSS).

#### 1.3: Construct arguments about geometric theorems using rigid transformations and/or logic.

G.CO.7: Construct arguments about lines and angles using theorems. 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. (Building upon standard in 8th grade Geometry.)

G.CO.9: Construct arguments about the relationships between two triangles using theorems. Theorems include: SSS, SAS, ASA, AAS, and HL.

G.CO.10: Construct arguments about parallelograms using theorems. 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. (Building upon prior knowledge in elementary and middle school.)

#### 1.4: Make geometric constructions.

G.CO.11: 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; 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.CO.12: Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.

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

#### 2.1: Understand similarity in terms of similarity transformations.

G.SRT.1: Use geometric constructions to verify the properties of dilations given by a center and a scale factor:

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

G.SRT.2: Recognize transformations as functions that take points in the plane as inputs and give other points as outputs and describe the effect of dilations on two-dimensional figures.

G.SRT.4: Understand the meaning of similarity for two-dimensional figures as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.

#### 2.2: Construct arguments about theorems involving similarity.

G.SRT.5: Construct arguments about triangles using theorems. 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, and AA.

G.SRT.6: Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.

#### 2.3: Define trigonometric ratios and solve problems involving right triangles.

G.SRT.7: Show 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.9: Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.

### G.C: Circles

#### 3.1: Understand and apply theorems about circles.

G.C.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.C.3: Construct arguments using properties of polygons inscribed and circumscribed about circles.

G.C.4: Construct inscribed and circumscribed circles for triangles.

G.C.5: Construct inscribed and circumscribed circles for polygons and tangent lines from a point outside a given circle to the circle.

#### 3.2: Find arc lengths and areas of sectors of circles.

G.C.6: 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

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

G.GPE.1: Write the equation of a circle given the center and radius or a graph of the circle; use the center and radius to graph the circle in the coordinate plane.

G.GPE.2: Derive the equation of a circle of given center and radius using the Pythagorean Theorem; graph the circle in the coordinate plane;

G.GPE.3: Complete the square to find the center and radius of a circle given by an equation.

G.GPE.4: Derive the equation of a parabola given a focus and directrix; graph the parabola in the coordinate plane.

G.GPE.5: Derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or difference of distances from the foci is constant; graph the ellipse or hyperbola in the coordinate plane.

#### 4.2: Use coordinates to prove simple geometric theorems algebraically.

G.GPE.8: Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, including the use of the distance and midpoint formulas.

### G.GMD: Geometric Measurement and Dimension

#### 5.1: Explain volume formulas and use them to solve problems.

G.GMD.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. Use dissection arguments and informal limit arguments.

G.GMD.2: Give an informal argument using Cavalieri’s principle for the formulas for the volume of a solid figure.

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.