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.

    • Lines, line segments, and rays (G-B.1)
    • 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)
    • Rotate polygons about a point (G-H.7)
    • Rotations: graph the image (G-H.8)
    • Rotations: find the coordinates (G-H.9)
    • Sequences of congruence transformations: graph the image (G-H.12)
    • Congruence transformations: mixed review (G-H.16)
    • Dilations: graph the image (G-I.1)
    • Dilations: find the coordinates (G-I.2)
    • Dilations: find the scale factor (G-I.4)
    • Dilations and parallel lines (G-I.6)

  • 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)
    • Draw lines of symmetry (G-J.3)
    • Count lines of symmetry (G-J.4)

  • 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)
    • Congruence transformations: mixed review (G-H.16)

• 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.

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

  • 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).

• 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)

• 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 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.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.

    • HSG.SRT.A.1.a 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 and parallel lines (G-I.6)

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

      • 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)

HSG.C Circles

• HSG.C.7 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.GPE Expressing Geometric Properties with Equations

• HSG.GPE.8 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.

    • 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)

• HSG.GPE.9 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.

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

HSG.MG Modeling with Geometry

• HSG.MG.10 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.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).