Mississippi

G-CO Congruence

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

  • G-CO.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.

    • Angle vocabulary (G-C.1)

  • G-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).

    • Classify congruence transformations (G-L.1)
    • Translations: graph the image (G-L.2)
    • Translations: find the coordinates (G-L.3)
    • Translations: write the rule (G-L.4)
    • Reflections: graph the image (G-L.5)
    • Reflections: find the coordinates (G-L.6)
    • Rotations: graph the image (G-L.8)
    • Rotations: find the coordinates (G-L.9)
    • Reflections and rotations: write the rule (G-L.)
    • Dilations: graph the image (G-L.15)
    • Dilations: find the coordinates (G-L.16)
    • Dilations: find the scale factor (G-L.18)

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

    • Transformations that carry a polygon onto itself (G-L.13)

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

    • Rotate polygons about a point (G-L.7)

  • G-CO.5 Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another.

    • Translations: graph the image (G-L.2)
    • Reflections: graph the image (G-L.5)
    • Rotations: graph the image (G-L.8)
    • Sequences of congruence transformations: graph the image (G-L.11)
    • Sequences of congruence transformations: find the rules (G-L.12)

  • Checkpoint opportunity

    • Checkpoint: Definitions of geometric objects (G-D.9)
    • Checkpoint: Transformations in the plane (G-L.21)
    • Checkpoint: Transformations of geometric figures (G-L.22)

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

  • G-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.

    • Sequences of congruence transformations: graph the image (G-L.11)
    • Transformations that carry a polygon onto itself (G-L.13)
    • Congruence transformations: mixed review (G-L.14)

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

  • Checkpoint opportunity

    • Checkpoint: Rigid motion and congruence (G-L.23)

• G-CO.C Prove geometric theorems

  • G-CO.9 Prove theorems about lines and angles.

    • Perpendicular Bisector Theorem (G-B.9)
    • Angle bisectors (G-C.6)
    • Proofs involving angles (G-C.9)
    • Proofs involving parallel lines I (G-D.7)
    • Proofs involving parallel lines II (G-D.8)

  • G-CO.10 Prove theorems about triangles.

    • Proofs involving isosceles triangles (G-K.10)
    • Construct the centroid or orthocenter of a triangle (G-M.7)
    • Find the centroid of a triangle (G-M.8)
    • Proofs involving triangles I (G-M.9)
    • Proofs involving triangles II (G-M.10)

  • G-CO.11 Prove theorems about parallelograms.

    • Properties of parallelograms (G-N.6)
    • Proving a quadrilateral is a parallelogram (G-N.7)
    • Properties of rhombuses (G-N.8)
    • Properties of squares and rectangles (G-N.9)

  • Checkpoint opportunity

    • Checkpoint: Line and angle theorems (G-D.10)
    • Checkpoint: Triangle theorems (G-M.11)
    • Checkpoint: Parallelogram theorems (G-N.16)

• G-CO.D Make geometric constructions

  • G-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.).

    • Construct a congruent segment (G-B.8)
    • Construct the midpoint or perpendicular bisector of a segment (G-B.15)
    • Construct an angle bisector (G-C.7)
    • Construct a congruent angle (G-C.8)
    • Construct a perpendicular line (G-D.2)
    • Construct parallel lines (G-D.6)
    • Construct an equilateral triangle or regular hexagon (G-G.5)
    • Construct a square (G-G.6)
    • Construct the circumcenter or incenter of a triangle (G-M.6)
    • Construct the centroid or orthocenter of a triangle (G-M.7)
    • Construct a tangent line to a circle (G-U.18)
    • Construct the inscribed or circumscribed circle of a triangle (G-U.22)

  • G-CO.13 Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.

    • Construct an equilateral triangle inscribed in a circle (G-U.19)
    • Construct a square inscribed in a circle (G-U.20)
    • Construct a regular hexagon inscribed in a circle (G-U.21)

  • Checkpoint opportunity

    • Checkpoint: Geometric constructions (G-U.26)

G-SRT Similarity, Right Triangles, and Trigonometry

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

  • G-SRT.1 Verify experimentally 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.

      • Dilations: graph the image (G-L.15)
      • Dilations and parallel lines (G-L.20)

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

      • Dilations: find length, perimeter, and area (G-L.17)
      • Dilations: find the scale factor (G-L.18)
      • Dilations: find the scale factor and center of the dilation (G-L.19)

  • G-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.

    • Identify similar figures (G-P.3)
    • Side lengths and angle measures in similar figures (G-P.4)
    • Similar triangles and similarity transformations (G-P.9)

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

    • Similarity rules for triangles (G-P.8)

  • Checkpoint opportunity

    • Checkpoint: Dilations (G-L.24)
    • Checkpoint: Similarity transformations (G-P.17)

• G-SRT.B Prove theorems involving similarity

  • G-SRT.4 Prove theorems about triangles.

    • Proofs involving triangles II (G-M.10)
    • Proofs involving similarity in right triangles (G-P.16)
    • Prove the Pythagorean theorem (G-Q.2)

  • G-SRT.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.

    • SSS and SAS Theorems (G-K.1)
    • Proving triangles congruent by SSS and SAS (G-K.2)
    • ASA and AAS Theorems (G-K.3)
    • Proving triangles congruent by ASA and AAS (G-K.4)
    • SSS, SAS, ASA, and AAS Theorems (G-K.5)
    • SSS Theorem in the coordinate plane (G-K.6)
    • Proving triangles congruent by SSS, SAS, ASA, and AAS (G-K.7)
    • Proofs involving corresponding parts of congruent triangles (G-K.8)
    • Congruency in isosceles and equilateral triangles (G-K.9)
    • Proofs involving isosceles triangles (G-K.10)
    • Hypotenuse-Leg Theorem (G-K.11)
    • Proofs involving triangles and quadrilaterals (G-N.14)
    • Similar triangles and indirect measurement (G-P.5)
    • Similarity rules for triangles (G-P.8)
    • Triangle Proportionality Theorem (G-P.11)
    • Similarity and altitudes in right triangles (G-P.12)
    • Prove similarity statements (G-P.14)
    • Prove proportions or angle congruences using similarity (G-P.15)

  • Checkpoint opportunity

    • Checkpoint: Triangle similarity and congruence (G-P.18)

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

  • G-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.

    • Trigonometric ratios: sin, cos, and tan (G-R.1)
    • Trigonometric ratios: csc, sec, and cot (G-R.2)
    • Trigonometric ratios in similar right triangles (G-R.3)

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

    • Sine and cosine of complementary angles (G-R.8)

  • G-SRT.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.

    • Pythagorean theorem (G-Q.1)
    • Converse of the Pythagorean theorem (G-Q.3)
    • Pythagorean Inequality Theorems (G-Q.4)
    • Trigonometric ratios: find a side length (G-R.10)
    • Trigonometric ratios: find an angle measure (G-R.11)
    • Solve a right triangle (G-R.12)

  • Checkpoint opportunity

    • Checkpoint: Right triangle trigonometry (G-R.18)

G-C Circles

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

  • G-C.1 Prove that all circles are similar.

    • Similarity of circles (G-P.10)

  • G-C.2 Identify and describe relationships among inscribed angles, radii, and chords.

    • Central angles and arc measures (G-U.2)
    • Arcs and chords (G-U.9)
    • Tangent lines (G-U.10)
    • Inscribed angles (G-U.12)
    • Angles in inscribed right triangles (G-U.13)
    • Angles formed by chords, secants, and tangents (G-U.16)
    • Segments formed by chords, secants, and tangents (G-U.17)

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

    • Construct the circumcenter or incenter of a triangle (G-M.6)
    • Angles in inscribed quadrilaterals I (G-U.14)
    • Angles in inscribed quadrilaterals II (G-U.15)
    • Construct the inscribed or circumscribed circle of a triangle (G-U.22)

  • Checkpoint opportunity

    • Checkpoint: Prove circles are similar (G-P.19)
    • Checkpoint: Angles and lines in circles (G-U.23)
    • Checkpoint: Inscribed and circumscribed circles (G-U.24)

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

  • G-C.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.

    • Understand arc length and sector area of a circle (G-U.3)
    • Arc length (G-U.4)
    • Convert between radians and degrees (G-U.5)
    • Radians and arc length (G-U.6)
    • Area of sectors (G-U.7)

  • Checkpoint opportunity

    • Checkpoint: Arc length and area of sectors (G-U.25)

G-GPE Expressing Geometric Properties with Equations

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

  • G-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.

    • Write equations of circles in standard form using properties (G-V.5)
    • Convert equations of circles from general to standard form (G-V.6)
    • Find properties of circles from equations in general form (G-V.7)
    • Graph circles from equations in standard form (G-V.8)
    • Graph circles from equations in general form (G-V.9)

  • Checkpoint opportunity

    • Checkpoint: Equations of circles (G-V.19)

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

  • G-GPE.4 Use coordinates to prove simple geometric theorems algebraically.

    • Classify shapes on the coordinate plane: justify your answer (G-N.13)
    • Determine if a point lies on a circle (G-V.3)

  • G-GPE.5 Prove 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).

    • Slopes of parallel and perpendicular lines (G-E.5)
    • Equations of parallel and perpendicular lines (G-E.6)
    • Find the distance between a point and a line (G-E.7)
    • Find the distance between two parallel lines (G-E.8)

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

    • Midpoints (G-B.6)
    • Midpoint formula: find the midpoint (G-B.10)
    • Partition a line segment in a given ratio (G-B.11)
    • Midpoint formula: find the endpoint (G-B.12)

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

    • Distance formula (G-B.13)
    • Area and perimeter in the coordinate plane I (G-S.6)
    • Area and perimeter in the coordinate plane II (G-S.7)

  • Checkpoint opportunity

    • Checkpoint: Partition a line segment (G-B.16)
    • Checkpoint: Coordinate proofs (G-E.9)
    • Checkpoint: Parallel and perpendicular lines (G-E.10)
    • Checkpoint: Area and perimeter in the coordinate plane (G-S.15)

G-GMD Geometric Measurement and Dimension

• G-GMD.A 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.

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

    • Volume of prisms and cylinders (G-T.5)
    • Volume of pyramids and cones (G-T.6)
    • Volume of spheres (G-T.7)

  • Checkpoint opportunity

    • Checkpoint: Volume (G-T.14)

• G-GMD.B Visualize relationships between two-dimensional and three-dimensional objects

  • G-GMD.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.

    • Cross sections of three-dimensional figures (G-H.4)
    • Solids of revolution (G-H.5)

  • Checkpoint opportunity

    • Checkpoint: Cross sections and solids of revolution (G-H.6)

G-MG Modeling with Geometry

• G-MG.A Apply geometric concepts in modeling situations

  • G-MG.1 Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).

    • Pythagorean theorem (G-Q.1)

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

    • Calculate density, mass, and volume (G-W.9)

  • G-MG.3 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).

  • Checkpoint opportunity

    • Checkpoint: Geometric modeling and design (G-W.10)
    • Checkpoint: Density (G-W.11)