Arkansas

HSG.CO Congruence

• HSG.CO.1 Investigate transformations in the plane

  • HSG.CO.A.1 Based on the undefined notions of point, line, plane, distance along a line, and distance around a circular arc, define: angle, line segment, circle, perpendicular lines, parallel lines.

    • Angle vocabulary (G-C.1)

  • HSG.CO.A.2 Represent transformations in the plane (e.g. using transparencies, tracing paper, geometry software, etc.). 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 dilation).

    • Classify congruence transformations (G-H.1)
    • Translations: graph the image (G-H.2)
    • Translations: find the coordinates (G-H.3)
    • Translations: write the rule (G-H.4)
    • Reflections: graph the image (G-H.5)
    • Reflections: find the coordinates (G-H.6)
    • Rotations: graph the image (G-H.8)
    • Rotations: find the coordinates (G-H.9)
    • Reflections and rotations: write the rule (G-H.10)
    • Dilations: graph the image (G-I.1)
    • Dilations: find the coordinates (G-I.2)
    • Dilations: find the scale factor (G-I.4)

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

    • Transformations that carry a polygon onto itself (G-H.15)

  • HSG.CO.A.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-H.7)

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

    • Translations: graph the image (G-H.2)
    • Reflections: graph the image (G-H.5)
    • Rotations: graph the image (G-H.8)
    • Sequences of congruence transformations: graph the image (G-H.12)
    • Sequences of congruence transformations: choose the sequence (G-H.13)
    • Sequences of congruence transformations: find the rules (G-H.14)

  • Checkpoint opportunity

    • Checkpoint: Definitions of geometric objects (G-D.9)
    • Checkpoint: Congruence transformations (G-H.17)
    • Checkpoint: Transformations in the plane (G-I.7)

• HSG.CO.2 Understand congruence in terms of rigid motions

  • HSG.CO.B.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-H.12)
    • Sequences of congruence transformations: choose the sequence (G-H.13)
    • Transformations that carry a polygon onto itself (G-H.15)
    • Congruence transformations: mixed review (G-H.16)
    • Determine if two figures are congruent: justify your answer (G-K.4)

  • HSG.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.

    • Prove triangles are congruent using rigid motions (G-L.1)

  • HSG.CO.B.8 Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions. Investigate congruence in terms of rigid motion to develop the criteria for triangle congruence (ASA, SAS, AAS, SSS, and HL).

  • Checkpoint opportunity

    • Checkpoint: Rigid motion and congruence (G-L.13)

• HSG.CO.3 Apply and prove geometric theorems

  • HSG.CO.C.9 Apply and prove theorems about lines and angles.

    • Perpendicular Bisector Theorem (G-B.9)
    • Midpoint formula: find the midpoint (G-B.10)
    • Distance formula (G-B.13)
    • Proofs involving angles (G-C.9)
    • Transversals: name angle pairs (G-D.3)
    • Transversals of parallel lines: find angle measures (G-D.4)
    • Proofs involving parallel lines I (G-D.7)
    • Proofs involving parallel lines II (G-D.8)

  • HSG.CO.C.10 Apply and prove theorems about triangles.

    • Triangle Angle-Sum Theorem (G-F.2)
    • Exterior Angle Theorem (G-F.3)
    • Exterior Angle Inequality (G-F.4)
    • Congruency in isosceles and equilateral triangles (G-L.10)
    • Proofs involving isosceles triangles (G-L.11)
    • Midsegments of triangles (G-N.1)
    • Triangles and bisectors (G-N.2)
    • Angle-side relationships in triangles (G-N.4)
    • Triangle Inequality Theorem (G-N.5)
    • Proofs involving triangles I (G-N.9)
    • Proofs involving triangles II (G-N.10)

  • HSG.CO.C.11 Apply and prove theorems about quadrilaterals.

    • Proving a quadrilateral is a parallelogram (G-O.7)
    • Proofs involving triangles and quadrilaterals (G-O.14)
    • Proofs involving quadrilaterals (G-O.15)

  • Checkpoint opportunity

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

• HSG.CO.4 Make geometric constructions

  • HSG.CO.D.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-N.6)
    • Construct the centroid or orthocenter of a triangle (G-N.7)
    • Construct a tangent line to a circle (G-X.1)
    • Construct the inscribed or circumscribed circle of a triangle (G-X.5)

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

    • Construct an equilateral triangle inscribed in a circle (G-X.2)
    • Construct a square inscribed in a circle (G-X.3)
    • Construct a regular hexagon inscribed in a circle (G-X.4)

  • Checkpoint opportunity

    • Checkpoint: Geometric constructions (G-X.6)

• HSG.CO.5 Logic and Reasoning

  • HSG.CO.E.14 Apply inductive reasoning and deductive reasoning for making predictions based on real world situations using: Conditional Statements (inverse, converse, and contrapositive); Venn Diagrams.

    • Identify hypotheses and conclusions (G-A.1)
    • Counterexamples (G-A.2)
    • Conditionals (G-A.3)
    • Converses, inverses, and contrapositives (G-A.5)
    • Biconditionals (G-A.6)

HSG.SRT Similarity, Right Triangles, and Trigonometry

• HSG.SRT.6 Understand similarity in terms of similarity transformations

  • HSG.SRT.A.1 Verify experimentally the properties of dilations given by a center and a scale factor. 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. The dilation of a line segment is longer or shorter in the ratio given by the scale factor.

    • Dilations: graph the image (G-I.1)
    • Dilations: find the coordinates (G-I.2)
    • Dilations: find the scale factor (G-I.4)

  • HSG.SRT.A.2 Given two figures: Use the definition of similarity in terms of similarity transformations to determine 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.

    • Side lengths and angle measures in similar figures (G-M.4)
    • Similar triangles and similarity transformations (G-M.9)
    • Similarity of circles (G-M.11)

  • HSG.SRT.A.3 Use the properties of similarity transformations to establish the AA, SAS~, SSS~ criteria for two triangles to be similar.

    • Similar triangles and similarity transformations (G-M.9)

  • Checkpoint opportunity

    • Checkpoint: Dilations (G-I.8)
    • Checkpoint: Similarity transformations (G-M.18)

• HSG.SRT.7 Apply and prove theorems involving similarity

  • HSG.SRT.B.4 Use triangle similarity to apply and prove theorems about triangles.

    • Similar triangles and indirect measurement (G-M.5)
    • Similarity rules for triangles (G-M.8)
    • Triangle Proportionality Theorem (G-M.12)
    • Similarity and altitudes in right triangles (G-M.13)
    • Prove similarity statements (G-M.15)
    • Prove proportions or angle congruences using similarity (G-M.16)
    • Proofs involving similarity in right triangles (G-M.17)
    • Proofs involving triangles II (G-N.10)
    • Prove the Pythagorean theorem (G-P.2)

  • HSG.SRT.B.5 Use congruence (SSS, SAS, ASA, AAS, and HL) and similarity (AA~, SSS~, SAS~) criteria for triangles to solve problems. Use congruence and similarity criteria to prove relationships in geometric figures.

    • Congruence statements and corresponding parts (G-K.1)
    • Solve problems involving corresponding parts (G-K.2)
    • Identify congruent figures (G-K.3)
    • SSS and SAS Theorems (G-L.2)
    • Proving triangles congruent by SSS and SAS (G-L.3)
    • ASA and AAS Theorems (G-L.4)
    • Proving triangles congruent by ASA and AAS (G-L.5)
    • SSS, SAS, ASA, and AAS Theorems (G-L.6)
    • SSS Theorem in the coordinate plane (G-L.7)
    • Proving triangles congruent by SSS, SAS, ASA, and AAS (G-L.8)
    • Proofs involving corresponding parts of congruent triangles (G-L.9)
    • Congruency in isosceles and equilateral triangles (G-L.10)
    • Hypotenuse-Leg Theorem (G-L.12)
    • Ratios in similar figures (G-M.1)
    • Similarity statements (G-M.2)
    • Identify similar figures (G-M.3)
    • Side lengths and angle measures in similar figures (G-M.4)
    • Similar triangles and indirect measurement (G-M.5)
    • Perimeters of similar figures (G-M.6)
    • Prove similarity statements (G-M.15)
    • Prove proportions or angle congruences using similarity (G-M.16)
    • Proofs involving similarity in right triangles (G-M.17)

  • Checkpoint opportunity

    • Checkpoint: Triangle similarity and congruence (G-M.19)

• HSG.SRT.8 Define trigonometric ratios and solve problems involving right triangles

  • HSG.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.

    • Trigonometric ratios: sin, cos, and tan (G-Q.1)
    • Trigonometric ratios with radicals: sin, cos, and tan (G-Q.2)
    • Trigonometric ratios: csc, sec, and cot (G-Q.3)
    • Trigonometric ratios in similar right triangles (G-Q.4)

  • HSG.SRT.C.7 Explain and use the relationship between the sine and cosine of complementary angles.

    • Sine and cosine of complementary angles (G-Q.9)

  • HSG.SRT.C.8 Use trigonometric ratios, special right triangles, and/or the Pythagorean Theorem to find unknown measurements of right triangles in applied problems.

    • Pythagorean theorem (G-P.1)
    • Converse of the Pythagorean theorem (G-P.3)
    • Pythagorean Inequality Theorems (G-P.4)
    • Trigonometric ratios: sin, cos, and tan (G-Q.1)
    • Trigonometric ratios with radicals: sin, cos, and tan (G-Q.2)
    • Trigonometric ratios: csc, sec, and cot (G-Q.3)
    • Trigonometric ratios in similar right triangles (G-Q.4)
    • Special right triangles (G-Q.5)
    • Trigonometric ratios: find a side length (G-Q.11)
    • Trigonometric ratios: find an angle measure (G-Q.12)
    • Solve a right triangle (G-Q.13)

  • Checkpoint opportunity

    • Checkpoint: Right triangle trigonometry (G)

• HSG.SRT.9 Apply trigonometric to general triangles

  • HSG.SRT.D.9 Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side.

    • Area of a triangle: sine formula (G-R.4)

  • HSG.SRT.D.10 Prove the Laws of Sines and Cosines and use them to solve problems.

    • Law of Sines (G-R.1)
    • Law of Cosines (G-R.2)
    • Area of a triangle: Law of Sines (G-R.5)

  • HSG.SRT.D.11 Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles.

    • Solve a triangle (G-R.3)

  • Checkpoint opportunity

    • Checkpoint: Laws of Sines and Cosines (G-R.6)

HSG.C Circles

• HSG.C.10 Understand and apply theorems about circles

  • HSG.C.A.1 Prove that all circles are similar.

    • Similarity of circles (G-M.11)

  • HSG.C.A.2 Identify, describe, and use relationships among angles, radii, segments, lines, arcs, and chords as related to circles.

    • Parts of a circle (G-W.1)
    • Central angles and arc measures (G-W.2)
    • Arc length (G-W.4)
    • Arcs and chords (G-W.9)
    • Tangent lines (G-W.10)
    • Inscribed angles (G-W.12)

  • HSG.C.A.3 Construct the inscribed and circumscribed circles of a triangle. Prove properties of angles for a quadrilateral inscribed in a circle.

    • Angles in inscribed right triangles (G-W.13)
    • Angles in inscribed quadrilaterals I (G-W.14)
    • Angles in inscribed quadrilaterals II (G-W.15)
    • Construct an equilateral triangle inscribed in a circle (G-X.2)
    • Construct the inscribed or circumscribed circle of a triangle (G-X.5)

  • Checkpoint opportunity

    • Checkpoint: Prove circles are similar (G-M.20)
    • Checkpoint: Angles and lines in circles (G-W.18)
    • Checkpoint: Inscribed and circumscribed circles (G-W.19)

• HSG.C.11 Find arc lengths and areas of sectors of circles

  • HSG.C.B.5 Derive using similarity that the length of the arc intercepted by an angle is proportional to the radius. Derive and use the formula for the area of a sector. Understand the radian measure of the angle as a unit of measure.

    • Arc length (G-W.4)
    • Convert between radians and degrees (G-W.5)
    • Radians and arc length (G-W.6)
    • Area of sectors (G-W.7)

  • Checkpoint opportunity

    • Checkpoint: Arc length and area of sectors (G-W.20)

HSG.GPE Expressing Geometric Properties with Equations

• HSG.GPE.12 Translate between the geometric description and the equation of a conic section

  • HSG.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.

    • Derive equations of circles using the Pythagorean theorem (G-Y.4)
    • Write equations of circles in standard form from graphs (G-Y.5)
    • Write equations of circles in standard form using properties (G-Y.6)
    • Convert equations of circles from general to standard form (G-Y.7)
    • Find properties of circles from equations in general form (G-Y.8)
    • Graph circles from equations in standard form (G-Y.9)
    • Graph circles from equations in general form (G-Y.10)

  • HSG.GPE.A.2 Derive the equation of a parabola given a focus and directrix.

    • Find the focus or directrix of a parabola (G-Z.2)
    • Write equations of parabolas in vertex form from graphs (G-Z.4)
    • Write equations of parabolas in vertex form using the focus and directrix (G-Z.5)
    • Write equations of parabolas in vertex form using properties (G-Z.6)
    • Find properties of a parabola from equations in general form (G-Z.8)
    • Graph parabolas (G-Z.9)

  • HSG.GPE.A.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.

    • Write equations of ellipses in standard form using properties (A2-FF.5)
    • Write equations of hyperbolas in standard form using properties (A2-GG.7)

  • Checkpoint opportunity

    • Checkpoint: Equations of circles (G-Y.11)
    • Checkpoint: Equations of parabolas (G-Z.10)

• HSG.GPE.13 Use coordinates to prove simple geometric theorems algebraically

  • HSG.GPE.B.4 Use coordinates to prove simple geometric theorems algebraically.

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

  • HSG.GPE.B.5 Prove the slope criteria for parallel and perpendicular lines. Use the slope criteria for parallel and perpendicular lines to solve geometric problems.

    • Slopes of parallel and perpendicular lines (G-E.4)
    • Equations of parallel and perpendicular lines (G-E.5)
    • Find the distance between a point and a line (G-E.6)
    • Find the distance between two parallel lines (G-E.7)

  • HSG.GPE.B.6 Find the midpoint between two given points; and find the endpoint of a line segment given the midpoint and one endpoint.

    • Midpoints (G-B.6)
    • Midpoint formula: find the midpoint (G-B.10)

  • HSG.GPE.B.7 Use coordinates to compute perimeters of polygons and areas of triangles and rectangles.

    • Distance formula (G-B.13)
    • Area and perimeter in the coordinate plane I (G-S.7)
    • Area and perimeter in the coordinate plane II (G-S.8)

  • Checkpoint opportunity

    • Checkpoint: Coordinate proofs (G-E.8)
    • Checkpoint: Parallel and perpendicular lines (G-E.9)
    • Checkpoint: Area and perimeter in the coordinate plane (G-S.16)

HSG.GMD Geometric measurement and dimension

• HSG.GMD.14 Explain volume formulas and use them to solve problems

  • HSG.GMD.A.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.

  • HSG.GMD.A.2 Give an informal argument using Cavalieri's principle for the formulas for the volume of a sphere and other solid figures.

  • HSG.GMD.A.3 Use volume formulas for cylinders, pyramids, cones, spheres, and to solve problems which may involve composite figures. Compute the effect on volume of changing one or more dimension(s).

    • Volume of prisms and cylinders (G-U.7)
    • Volume of pyramids and cones (G-U.8)
    • Volume of spheres (G-U.9)

  • Checkpoint opportunity

    • Checkpoint: Volume (G-U.17)

• HSG.GMD.15 Visualize relationships between two-dimensional and three-dimensional objects

  • HSG.GMD.B.4 Identify the shapes of two-dimensional cross-sections of three- dimensional objects. Identify three-dimensional objects generated by rotations of two-dimensional objects.

    • Parts of three-dimensional figures (G-T.1)
    • Cross sections of three-dimensional figures (G-T.4)
    • Solids of revolution (G-T.5)

  • Checkpoint opportunity

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

HSG.MG Modeling with Geometry

• HSG.MG.16 Apply geometric concepts in modeling situations

  • HSG.MG.A.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-P.1)

  • HSG.MG.A.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-V.9)

  • HSG.MG.A.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-V.10)
    • Checkpoint: Density (G-V.11)