### OH.Math.HSG.CO: Congruence

#### OH.Math.HSG.CO.A: Experiment with transformations in the plane.

OH.Math.HSG.CO.1: Know precise definitions of ray, angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and arc length.

OH.Math.HSG.CO.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.

OH.Math.HSG.CO.3: Identify the symmetries of a figure, which are the rotations and reflections that carry it onto itself.

OH.Math.HSG.CO.3a: Identify figures that have line symmetry; draw and use lines of symmetry to analyze properties of shapes.

OH.Math.HSG.CO.3b: Identify figures that have rotational symmetry; determine the angle of rotation, and use rotational symmetry to analyze properties of shapes.

OH.Math.HSG.CO.4: Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.

OH.Math.HSG.CO.5: Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using items such as graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another.

#### OH.Math.HSG.CO.B: Understand congruence in terms of rigid motions.

OH.Math.HSG.CO.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.

OH.Math.HSG.CO.8: Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.

#### OH.Math.HSG.CO.C: Prove geometric theorems both formally and informally using a variety of methods.

OH.Math.HSG.CO.9: Prove and apply theorems about lines and angles.

1.3.1.1: Theorems include but are not restricted to the following: 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.

OH.Math.HSG.CO.10: Prove and apply theorems about triangles.

1.3.2.1: Theorems include but are not restricted to the following: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles 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.

OH.Math.HSG.CO.11: Prove and apply theorems about parallelograms.

1.3.3.1: Theorems include but are not restricted to the following: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals.

#### OH.Math.HSG.CO.D: Make geometric constructions.

OH.Math.HSG.CO.12: Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.).

1.4.1.1: 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.

OH.Math.HSG.CO.13: Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.

#### OH.Math.HSG.CO.E: Classify and analyze geometric figures.

OH.Math.HSG.CO.14: Classify two-dimensional figures in a hierarchy based on properties.

### OH.Math.HSG.SRT: Similarity, Right Triangles, and Trigonometry

#### OH.Math.HSG.SRT.A: Understand similarity in terms of similarity transformations.

OH.Math.HSG.SRT.1: Verify experimentally the properties of dilations given by a center and a scale factor:

OH.Math.HSG.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.

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

OH.Math.HSG.SRT.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.

OH.Math.HSG.SRT.3: Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.

#### OH.Math.HSG.SRT.B: Prove and apply theorems both formally and informally involving similarity using a variety of methods.

OH.Math.HSG.SRT.4: Prove and apply theorems about triangles.

2.2.1.1: Theorems include but are not restricted to the following: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity.

OH.Math.HSG.SRT.5: Use congruence and similarity criteria for triangles to solve problems and to justify relationships in geometric figures that can be decomposed into triangles.

#### OH.Math.HSG.SRT.C: Define trigonometric ratios, and solve problems involving right triangles.

OH.Math.HSG.SRT.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.

OH.Math.HSG.SRT.8: Solve problems involving right triangles.

OH.Math.HSG.SRT.8a: Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems if one of the two acute angles and a side length is given.

OH.Math.HSG.SRT.8b: Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.

### OH.Math.HSG.C: Circles

#### OH.Math.HSG.C.A: Understand and apply theorems about circles.

OH.Math.HSG.C.2: Identify and describe relationships among angles, radii, chords, tangents, and arcs and use them to solve problems.

3.1.2.1: Include the relationship between central, inscribed, and circumscribed angles and their intercepted arcs; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle.

OH.Math.HSG.C.3: Construct the inscribed and circumscribed circles of a triangle; prove and apply the property that opposite angles are supplementary for a quadrilateral inscribed in a circle.

#### OH.Math.HSG.C.B: Find arc lengths and areas of sectors of circles.

OH.Math.HSG.C.5: Find arc lengths and areas of sectors of circles.

OH.Math.HSG.C.5a: Apply similarity to relate the length of an arc intercepted by a central angle to the radius. Use the relationship to solve problems.

OH.Math.HSG.C.5b: Derive the formula for the area of a sector, and use it to solve problems.

OH.Math.HSG.C.6: Derive formulas that relate degrees and radians, and convert between the two.

### OH.Math.HSG.GPE: Expressing Geometric Properties with Equations

#### OH.Math.HSG.GPE.A: Translate between the geometric description and the equation for a conic section.

OH.Math.HSG.GPE.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.

OH.Math.HSG.GPE.2: Derive the equation of a parabola given a focus and directrix.

OH.Math.HSG.GPE.3: 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.

#### OH.Math.HSG.GPE.B: Use coordinates to prove simple geometric theorems algebraically and to verify specific geometric statements.

OH.Math.HSG.GPE.4: Use coordinates to prove simple geometric theorems algebraically and to verify geometric relationships algebraically, including properties of special triangles, quadrilaterals, and circles.

OH.Math.HSG.GPE.7: Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.

### OH.Math.HSG.GMD: Geometric Measurement and Dimension

#### OH.Math.HSG.GMD.A: Explain volume formulas, and use them to solve problems.

OH.Math.HSG.GMD.1: 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.

5.1.1.1: Use dissection arguments, Cavalieri's principle, and informal limit arguments.

OH.Math.HSG.GMD.2: Give an informal argument using Cavalieri’s principle for the formulas for the volume of a sphere and other solid figures.

OH.Math.HSG.GMD.3: Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.

#### OH.Math.HSG.GMD.C: Understand the relationships between lengths, area, and volumes.

OH.Math.HSG.GMD.6: When figures are similar, understand and apply the fact that when a figure is scaled by a factor of k, the effect on lengths, areas, and volumes is that they are multiplied by k, k², and k³, respectively.

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.