# Academic Standard

Geometry and Measurement
Initiative:
Tennessee Diploma Project
Set:
Mathematics
Type:
Standard
Code:
4
Grade range:
9 to 12
Conceptual StrandGeometry studies geometric shapes and structures and their characteristics and relationships. Understanding what a measurable attribute is and becoming familiar with the units and processes that are used in measuring attributes are the major emphasis of this standard.Guiding QuestionHow can students make and explore conjectures about geometry and reason carefully about geometric ideas while expanding their understanding of precision in measurement?

Elements within this Standard

Course Level Expectation
Develop the structures of geometry, such as lines, angles, planes, and planar figures, and explore their properties and relationships.
Describe the properties of regular polygons, including comparative classification of them and special points and segments.
Develop an understanding of the tools of logic and proof, including aspects of formal logic as well as construction of proofs.
Develop geometric intuition and visualization through performing geometric constructions with straightedge/compass and with technology.
Extend the study of planar figures to three-dimensions, including the classical solid figures, and develop analysis through cross-sections.
Generate formulas for perimeter, area, and volume, including their use, dimensional analysis, and applications.
Apply the major concepts of transformation geometry to analyzing geometric objects and symmetry.
Establish processes for determining congruence and similarity of figures, especially as related to scale factor, contextual applications, and transformations.
Develop the role of circles in geometry, including angle measurement, properties as a geometric figure, and aspects relating to the coordinate plane.
Develop the tools of right triangle trigonometry in the contextual applications, including the Pythagorean Theorem, Law of Sines and Law of Cosines.
Check For Understanding
Recognize that there are geometries, other than Euclidean geometry, in which the parallel postulate is not true and discuss unique properties of each.
Compare and contrast inductive reasoning and deductive reasoning for making predictions and valid conclusions based on contextual situations.
Solve problems involving betweeness of points and distance between points (including segment addition).
Describe and recognize minimal conditions necessary to define geometric objects.
Use vertical, adjacent, complementary, and supplementary angle pairs to solve problems and write proofs.
Describe the intersection of lines (in the plane and in space), a line and a plane, or of two planes.
Identify perpendicular planes, parallel planes, a line parallel to a plane, skew lines, and a line perpendicular to a plane.
Apply properties and theorems about angles associated with parallel and perpendicular lines to solve problems.
Classify triangles, quadrilaterals, and polygons (regular, non-regular, convex and concave) using their properties.
Identify and apply properties and relationships of special figures (e.g., isosceles and equilateral triangles, family of quadrilaterals, polygons, and solids).
Use the triangle inequality theorems (e.g., Exterior Angle Inequality Theorem, Hinge Theorem, SSS Inequality Theorem, Triangle Inequality Theorem) to solve
Apply the Angle Sum Theorem for polygons to find interior and exterior angle measures given the number of sides, to find the number of sides given angle
Locate, describe, and draw a locus in a plane or space (e.g., fixed distance from a point on a plane, fixed distance from a point in space, fixed distance from
Identify and use medians, midsegments, altitudes, angle bisectors, and perpendicular bisectors of triangles to solve problems (e.g., find segment lengths, angle
Identify, write, and interpret conditional and bi-conditional statements along with the converse, inverse, and contra-positive of a conditional statement.
Analyze and create truth tables to evaluate conjunctions, disjunctions, conditionals, inverses, contra-positives, and bi-conditionals.
Use the Law of Detachment, Law of Syllogism, conditional statements, and bi-conditional statements to draw conclusions.
Use counterexamples, when appropriate, to disprove a statement.
Use coordinate geometry to prove properties of plane figures.
Prove key basic theorems in geometry (i.e., Pythagorean Theorem, the sum of the angles of a triangle is 180 degrees, characteristics of quadrilaterals, and the
Use properties of and theorems about parallel lines, perpendicular lines, and angles to prove basic theorems in Euclidean geometry (e.g., two lines parallel to
Perform basic geometric constructions using a straight edge and a compass, paper folding, graphing calculator programs, and computer software packages (i.e.,
Describe the polyhedron or solid that can be made from a given net including the Platonic Solids.
Develop and use special formulas relating to polyhedra (e.g., Eulers Formula).
Use properties of prisms, pyramids, cylinders, cones, spheres, and hemispheres to solve problems.
Describe and draw cross-sections (including the conic sections) of prisms, cylinders, pyramids, spheres, and cones.
Use right triangle trigonometry to find the area and perimeter of quadrilaterals (e.g. square, rectangle, rhombus, parallelogram, trapezoid, and kite).
Derive and use the formulas for the area and perimeter of a regular polygon. (A=1/2 ap)
Extend the effect of a scale factor k in similar objects to include the impact on volume calculations and transformations.
Use right triangle relationships or the Pythagorean Theorem to find the lateral area (if possible), surface area, and volume of prisms, cylinders, cones,
Use properties of single transformations and compositions of transformations to determine their effect on geometric figures (e.g. reflections across lines of
Recognize, identify and apply types of symmetries (point, line, rotational) of two- and three- dimensional figures.
Use transformations to create and analyze tessellations and investigate the use of tessellations in architecture, mosaics, and artwork.
Create and analyze geometric designs using rigid motions (compositions of reflections, translations, and rotations).
Prove that two triangles are congruent by applying the SSS, SAS, ASA, AAS, and HL congruence statements.
Use several methods, including AA, SSS, and SAS, to prove that two triangles are similar.
Identify similar figures and use ratios and proportions to solve mathematical and real-world problems (e.g., Golden Ratio).
Use the principle that corresponding parts of congruent triangles are congruent to solve problems.
Identify lines and line segments associated with circles.
Find angle measures, intercepted arc measures, and segment lengths formed by radii, chords, secants, and tangents intersecting inside and outside circles.
Use inscribed and circumscribed polygons to solve problems concerning segment length and angle measures.
Use geometric mean to solve problems involving relationships that exist when the altitude is drawn to the hypotenuse of a right triangle.
Apply the Pythagorean Theorem and its converse to triangles to solve mathematical and contextual problems in two- or three-dimensional situations.
Identify and use Pythagorean triples in right triangles to find lengths of an unknown side in two- or three-dimensional situations.
Use the converse of the Pythagorean Theorem to classify a triangle by its angles (right, acute, or obtuse).
Apply properties of 30 - 60 - 90 and 45 - 45 - 90 to determine side lengths of triangles.
Find the sine, cosine and tangent ratios of an acute angle of a right triangle given the side lengths.
Define, illustrate, and apply angles of elevation and angles of depression in real-world situations.
Use the Law of Sines (excluding the ambiguous case) and the Law of Cosines to find missing side lengths and/or angle measures in non-right triangles.
State Performance Indicator
Differentiate between Euclidean and non-Euclidean geometries.
Define, identify, describe, and/or model plane figures using appropriate mathematical symbols (including collinear and non-collinear points, lines, segments,
Identify, describe and/or apply the relationships and theorems involving different types of triangles, quadrilaterals and other polygons.
Analyze different types and formats of proofs.
Describe solids and/or surfaces in three-dimensional space when given two-dimensional representations for the surfaces of three-dimensional objects.
Use various area of triangle formulas to solve contextual problems (e.g., Herons formula, the area formula for an equilateral triangle, and A = ab sin C).
Compute the area and/or perimeter of triangles, quadrilaterals and other polygons when one or more additional steps are required (e.g. find missing dimensions
Solve problems involving area, circumference, area of a sector, and/or arclength of a circle.
Use right triangle trigonometry and cross-sections to solve problems involving surface areas and/or volumes of solids.
Identify, describe, and/or apply transformations on two and three dimensional geometric shapes.
Use basic theorems about similar and congruent triangles to solve problems.
Solve problems involving congruence, similarity, proportional reasoning and/or scale factor of two similar figures or solids.
Identify, analyze and/or use basic properties and theorems of circles to solve problems (including those relating right triangles and circles).
Use properties of right triangles to solve problems (such as involving the relationship formed when the altitude to the hypotenuse of a right triangle is drawn).
Determine and use the appropriate trigonometric ratio for a right triangle to solve a contextual problem.