High School — Geometry

Other Ohio Mathematics sets

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• High School — Algebra
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• High School — Number and Quantity

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.G.CO.1

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.2

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

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

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.G.CO.5

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.6

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.7

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

Prove and apply theorems about lines and angles. 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.G.CO.9

Prove and apply theorems about triangles. 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.G.CO.10

Prove and apply theorems about parallelograms.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.CO.13

Classify two-dimensional figures in a hierarchy based on properties.G.CO.14

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

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.2

Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.G.SRT.3

Prove and apply theorems about triangles.G.SRT.4

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

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.6

Explain and use the relationship between the sine and cosine of complementary angles.G.SRT.7

Solve problems involving right triangles.G.SRT.8

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.(+)G.SRT.9

Explain proofs of the Laws of Sines and Cosines and use the Laws to solve problems.(+)G.SRT.10

Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles, e.g., surveying problems, resultant forces.(+)G.SRT.11

Prove that all circles are similar using transformational arguments.G.C.1

Identify and describe relationships among angles, radii, chords, tangents, and arcs and use them to solve problems. 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.G.C.2

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.G.C.3

Construct a tangent line from a point outside a given circle to the circle.(+)G.C.4

Find arc lengths and areas of sectors of circles.G.C.5

Derive formulas that relate degrees and radians, and convert between the two.G.C.6

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.GPE.1

Derive the equation of a parabola given a focus and directrix.(+)G.GPE.2

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.(+)G.GPE.3

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

Justify 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.5

Find the point on a directed line segment between two given points that partitions the segment in a given ratio.G.GPE.6

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

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. Use dissection arguments, Cavalieri's principle, and informal limit arguments.G.GMD.1

Give an informal argument using Cavalieri's principle for the formulas for the volume of a sphere and other solid figures.(+)G.GMD.2

Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.G.GMD.3

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.GMD.4

Understand how and when changes to the measures of a figure (lengths or angles) result in similar and non-similar figures.G.GMD.5

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.G.GMD.6

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.1

Apply concepts of density based on area and volume in modeling situations, e.g., persons per square mile, BTUs per cubic foot.G.MG.2

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.G.MG.3