Grades: 8 - 12
Credit: 1
Prerequisite: Algebra 1

Students investigate properties of triangles, quadrilaterals, polygons, circles, and solids using inductive and deductive reasoning. Conjectures about properties and relationships are developed inductively and then verified deductively. Vectors, transformations, algebra, and technologies are used as tools to solve geometry problems. Items marked with ® are optional enrichment topics.

Indicators marked with an asterisk * relate directly to the Virginia Standards of Learning for Geometry. All Virginia Standards of Learning for Geometry are included in this course

Benchmarks and indicators are organized by the following strands: 

I. Fundamental Concepts

II. Transformations and Symmetry

III. Triangles

IV. Coordinates and Vectors

V. Quadrilaterals

VI. Polygons

VII. Circles

VIII. Solids

IX. Logic and Proof

X. Geometries®

Benchmark
Students examine concepts fundamental to the study of geometry in two and three dimensions. Through investigations they draw conclusions about relationships involving points, lines, planes, and angles.

1. Identify, draw, and label points, lines, planes, line segments, rays, and angles in space, using correct symbolic notation.
2. Investigate point, line, and plane relationships, including collinear, coplanar, parallel, perpendicular, and skew, in two- and three-dimensional settings.
3. Investigate angle measures for: angles formed by intersecting lines; complementary and supplementary angles; and angles formed by a transversal cutting a system of lines.*
4. Construct the following, using only a compass and straightedge: congruent line segments, segment bisector, a perpendicular to a line from a point not on the line, a perpendicular to a line from a point on the line, angle bisector, and congruent angles.*

Benchmark
Students investigate and apply symmetries, isometries, and dilations through geometric movement.

1. Apply vocabulary and symbolic notation for mappings and various transformations.
2. Create and compare preimages and images for reflections, translations, glide reflections, rotations, and dilations on plane figures.*
3. Relate isometries to congruence and dilations to similarity. ®
4. Create and compare preimages and images for compositions of transformations of plane figures.
5. Describe and identify line, point, rotational, and translational symmetry of plane figures.*
6. Explore patterns and angle relationships with polygons using tessellations and symmetry.*

Benchmark
Students identify properties and investigate relationships of triangles using symmetry, congruence, and transformation.

1. Classify and compare triangles by definition and properties, including inequality properties.*
2. Investigate necessary and sufficient conditions to determine a unique triangle.*
3. Describe and explore symmetries in triangles.
4. Identify and investigate medians, altitudes, perpendicular bisectors of sides, angle bisectors, and transversals parallel to one side of a triangle.
5. Construct the medians, altitudes, perpendicular bisectors, and angle bisectors and their respective points of concurrency in a triangle. ®
6. Compare corresponding angles and corresponding sides of triangles to determine similar and congruent triangles.*
7. Determine perimeter and area of triangles.
8. Use proportional reasoning to solve practical problems involving similar triangles.*
9. Solve problems involving the lengths of sides in right triangles, including 45-45-90 and 30-60-90 triangles. *
10. Define trigonometric ratios and use them to solve real-life applications.*

Benchmark
Students derive and apply formulas using coordinate geometry. Vectors and operations on vectors are used as a tool for exploring geometric concepts.

1. Informally derive and apply formulas for slope, midpoint, and distance.*
2. Find distance and midpoints in three space. ®
3. Compare the slopes of lines with their relative positions in a coordinate plane, including parallel, perpendicular, horizontal, and vertical.
4. Identify, draw, and label vectors using appropriate notation in two- (and three-®) space representations.*
5. Express the addition and scalar multiplication of vectors both graphically and algebraically.*
6. Represent vectors as matrices and determine resultants by matrix addition.*
7. Apply vectors to practical situations.*

Benchmark
Students investigate properties and relationships of quadrilaterals using symmetry, congruence, and transformations.

1. Examine properties of angles, sides, and diagonals of quadrilaterals and determine measures using algebraic and coordinate methods.*
2. Compare, contrast, and classify quadrilaterals.
3. Determine perimeter and area measures of quadrilaterals and apply these to real-life situations.*
4. Solve problems involving the ratios of sides, perimeters, and areas of similar quadrilaterals.*

Benchmark
Students investigate a variety of polygons to determine measures and properties, including fundamental relationships between angles and segments.

1. Identify and classify different types of polygons.
2. Determine relationships between measures of angles (interior and exterior) and number of sides of different polygons.*
3. Informally derive and apply formulas for the area of regular polygons.
4. Find perimeters and areas of figures that can be divided into polygonal and circular regions. ®

Benchmark
Students investigate properties of circles focusing on relationships among angles, arcs, lines, and chords.

1. Use appropriate vocabulary to identify and describe circles and their related parts.
2. Apply relationships among tangents, radii, and chords to solve problems involving measurement in circles.*
3. Determine and apply relationships between arcs and angles determined by radii, chords, secants, and tangents.*
4. Determine circumference, arc length, and areas of circles, sectors, and segments.*

Students investigate polyhedrons by extending concepts of plane geometry to three space.

1. Sketch polyhedrons and identify relevant parts.*
2. Draw and recognize nets of various polyhedrons in order to make three dimensional models.*
3. Investigate right triangles in cones, pyramids, and spheres.
4. Informally derive and use formulas for determining surface area and volume of a variety of nonoblique solids and apply these in real-world situations.*
5. Investigate ratios of areas and volumes of similar solids.*
6. Use and create models and representations of practical situations, which include scale drawings, perspective drawings, blueprints, or computer simulations.*
7. Explore results of oblique cuts on traditional solids. ®

Benchmark
Students apply several alternative methods for producing deductive arguments to verify hypotheses.

1. Use inductive and deductive reasoning, including the law of syllogism, as a means for developing logical arguments.*
2. Create, discuss and compare conditional statements and their inverses, converses, and contrapositives and translate statements into symbolic form.*
3. Disprove a conjecture by producing a counterexample to a proposed hypothesis.
4. Investigate coordinate, two-column, flow, paragraph, and indirect proofs as well as verbal argument as appropriate methods for producing logical deductive arguments.*
5. Write deductive arguments to prove or disprove a hypothesis. (Students choose their own methods for producing each proof)*
6. Use Venn diagrams to illustrate logical arguments involving quantifiers.*

X. Geometries®

Benchmark
Students investigate non-Euclidean geometries and other geometry-related topics, including topology, knot theory, networks, locus, and fractals.

1. Investigate alternatives to Euclidís parallel postulate. ®
2. Compare properties and relationships of geometries of flat surfaces, spherical surfaces, and pseudospherical surfaces. ®
3. Investigate topological surfaces. ®
4. Find and construct the locus of points in a plane and in space and use loci to solve real-life problems. ®
5. Identify and create fractals. ®
6. Determine characteristics that make networks traversable. ®

All students will take the Virginia Standards of Learning Test for Geometry.

Last update: September 10, 1998