Arizona Geometry Math Standards

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G.N Number and Quantity

• G.N-Q Quantities
  • G.N-Q.A Reason quantitatively and use units to solve problems.
    • G.N-Q.A.1 Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays, include utilizing real-world context.
    • G.N-Q.A.2 Define appropriate quantities for the purpose of descriptive modeling. Include problem-solving opportunities utilizing real-world context.
    • G.N-Q.A.3 Choose a level of accuracy appropriate to limitations on measurement when reporting quantities utilizing real-world context.

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.
    • Checkpoint opportunity
  • 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.7 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.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.
    • Checkpoint opportunity
  • 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.
    • Checkpoint opportunity
  • 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.
    • Checkpoint opportunity
• 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.1.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.G-SRT.A.1.b 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.
    • Checkpoint opportunity
  • 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.
    • Checkpoint opportunity
  • 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.7 Explain and use the relationship between the sine and cosine of complementary 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.
    • Checkpoint opportunity
• G.G-C Circles
  • G.G-C.A Understand and apply theorems about circles.
    • G.G-C.A.1 Prove that all circles are similar.
    • 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.
    • Checkpoint opportunity
  • 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.
    • Checkpoint opportunity
• 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.
    • Checkpoint opportunity
  • G.G-GPE.B Use coordinates to prove geometric theorems algebraically.
    • G.G-GPE.B.4 Use coordinates to algebraically prove or disprove geometric relationships algebraically. Relationships include: proving or disproving geometric figures given specific points in the coordinate plane; and proving or disproving if a specific point lies on a given circle.
    • G.G-GPE.B.5 Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems, including finding the equation of a line parallel or perpendicular to a given line that passes through a given point.
    • G.G-GPE.B.6 Find the point on a directed line segment between two given points that partitions the segment in a given ratio.
    • G.G-GPE.B.7 Use coordinates to compute perimeters of polygons and areas of triangles and rectangles.
    • Checkpoint opportunity
• 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.
    • Checkpoint opportunity
  • G.G-GMD.B Visualize relationships between two-dimensional and three-dimensional objects.
    • G.G-GMD.B.4 Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects.
    • Checkpoint opportunity
• G.G-MG Modeling with Geometry
  • G.G-MG-A Apply geometric concepts in modeling situations.
    • G.G-MG.A.1 Use geometric shapes, their measures, and their properties to describe objects utilizing real-world context.
    • G.G-MG.A.2 Apply concepts of density based on area and volume in modeling situations utilizing real-world context.
    • G.G-MG.A.3 Apply geometric methods to solve design problems utilizing real-world context.
    • Checkpoint opportunity