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Geometry

Geometry

Proof-based high school Geometry, from congruence to circles, with every topic tied to the SAT.

Category
Math and computer science
Units
11 units
Exam
Exam May 24, 2027 (in 239 days)
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Unit 1 is free with an account. No card, no trial clock.

What the course covers

A full honors high school Geometry course aligned to the Texas TEKS (§111.41) and built at Pre-AP rigor. It runs from coordinate tools and logic through transformations, congruence, similarity, right-triangle trigonometry, polygons, circles, measurement in two and three dimensions, and probability. Students write real proofs (two-column, paragraph, and flow), model authentic situations, and move between diagrams, coordinates, and equations. Every item that feeds the Digital SAT carries its SAT domain, so the course doubles as preparation for the SAT Geometry and Trigonometry questions and the probability and data questions.

11 units, with exam weights

  1. Unit 1

    Foundations and Coordinate Geometry

    6-8% of examFree
    The vocabulary and measurement tools the rest of the course assumes: points, lines, and planes; distance and midpoint on a number line and in the plane; dividing a segment in a ratio; angle measure; angle pairs; and the addition postulates that turn a diagram into an equation.
    6 topics
    1. 1.1Points, Lines, and Planes
    2. 1.2Segments, Distance, and the Distance Formula
    3. 1.3Midpoint and Partitioning Segments
    4. 1.4Angles and Angle Measure
    5. 1.5Angle Pairs
    6. 1.6Segment and Angle Addition; Bisectors

  2. Unit 2

    Reasoning, Logic, and Proof

    6-8% of exam
    How mathematicians know things. Inductive conjectures and counterexamples, conditional statements and their related forms, valid deductive arguments, algebraic proof, the three proof formats, and the classical compass-and-straightedge constructions.
    6 topics
    1. 2.1Inductive Reasoning and Conjectures
    2. 2.2Conditional Statements
    3. 2.3Deductive Reasoning (Law of Detachment and Law of Syllogism)
    4. 2.4Algebraic Proof and Properties of Equality
    5. 2.5Two-Column, Paragraph, and Flow Proofs
    6. 2.6Constructions with Compass and Straightedge

  3. Unit 3

    Parallel and Perpendicular Lines

    7-9% of exam
    Angles formed when a transversal crosses lines, proving lines parallel, slope criteria for parallel and perpendicular lines, equations of lines, the distance from a point to a line, perpendicular bisectors, and how parallel lines behave in spherical geometry.
    6 topics
    1. 3.1Lines and Transversals; Angle Pairs
    2. 3.2Proving Lines Parallel
    3. 3.3Slope, and Slopes of Parallel and Perpendicular Lines
    4. 3.4Equations of Lines (Slope-Intercept, Point-Slope, Standard)
    5. 3.5Distance from a Point to a Line; Perpendicular Bisectors
    6. 3.6Euclidean and Spherical Geometry

  4. Unit 4

    Transformations and Symmetry

    7-9% of exam
    Translations, reflections, rotations, and dilations in coordinate notation; compositions and the sequence that maps one figure onto another; reflectional and rotational symmetry; congruence as rigid motion and similarity as dilation.
    5 topics
    1. 4.1Translations
    2. 4.2Reflections
    3. 4.3Rotations
    4. 4.4Compositions of Transformations and Symmetry
    5. 4.5Dilations and Scale Factor

  5. Unit 5

    Triangles and Congruence

    11-13% of exam
    Triangle angle theorems, congruence as rigid motion, the five congruence criteria and CPCTC, isosceles and equilateral triangles, special segments and their points of concurrency, and the triangle inequalities.
    6 topics
    1. 5.1Triangle Angle Sum and Exterior Angle Theorems; Classifying Triangles
    2. 5.2Congruent Figures and CPCTC
    3. 5.3Proving Triangles Congruent (SSS, SAS, ASA, AAS, HL)
    4. 5.4Isosceles and Equilateral Triangles
    5. 5.5Special Segments and Points of Concurrency
    6. 5.6Triangle Inequalities

  6. Unit 6

    Similarity

    9-11% of exam
    Ratios and proportions, similar polygons defined through dilations, the AA, SSS, and SAS similarity criteria, proportional segments and midsegments, and the geometric-mean relationships created by the altitude to a hypotenuse.
    5 topics
    1. 6.1Ratios, Proportions, and Scale
    2. 6.2Similar Polygons
    3. 6.3Proving Triangles Similar (AA, SSS, SAS)
    4. 6.4Proportional Segments (Triangle Proportionality and Midsegments)
    5. 6.5Geometric Mean and Right-Triangle Altitude Relationships

  7. Unit 7

    Right Triangles and Trigonometry

    11-13% of exam
    The Pythagorean theorem and its converse, 45-45-90 and 30-60-90 triangles, the sine, cosine, and tangent ratios, solving right triangles with inverse trigonometry, angles of elevation and depression, and the laws of sines and cosines.
    6 topics
    1. 7.1The Pythagorean Theorem and Its Converse
    2. 7.2Special Right Triangles (45-45-90 and 30-60-90)
    3. 7.3Trigonometric Ratios (Sine, Cosine, Tangent)
    4. 7.4Solving Right Triangles and Inverse Trigonometry
    5. 7.5Angles of Elevation and Depression
    6. 7.6Law of Sines and Law of Cosines

  8. Unit 8

    Quadrilaterals and Polygons

    7-9% of exam
    Interior and exterior angle sums of polygons, properties of parallelograms, rectangles, rhombi, squares, trapezoids, and kites, proving a quadrilateral is a particular type, and coordinate proofs.
    5 topics
    1. 8.1Angles of Polygons (Interior and Exterior Sums)
    2. 8.2Parallelograms and Their Properties
    3. 8.3Rectangles, Rhombi, and Squares
    4. 8.4Trapezoids and Kites
    5. 8.5Coordinate Proofs with Quadrilaterals

  9. Unit 9

    Circles

    9-11% of exam
    Central angles and arcs, arc length, sector area, and radian measure, chords, inscribed angles and cyclic quadrilaterals, tangents and secants with their angle and segment relationships, and equations of circles.
    6 topics
    1. 9.1Circle Vocabulary; Central Angles and Arcs
    2. 9.2Arc Length and Sector Area
    3. 9.3Chords and Their Properties
    4. 9.4Inscribed Angles and Quadrilaterals
    5. 9.5Tangents, Secants, and Angle and Segment Relationships
    6. 9.6Equations of Circles

  10. Unit 10

    Area, Surface Area, and Volume

    10-12% of exam
    Areas of triangles, quadrilaterals, regular polygons, and composite figures, surface area and volume of prisms, cylinders, pyramids, cones, and spheres, cross-sections and solids of revolution, and how changing dimensions changes area and volume.
    6 topics
    1. 10.1Area of Triangles and Quadrilaterals
    2. 10.2Area of Regular Polygons and Composite Figures
    3. 10.3Surface Area of Prisms, Cylinders, Pyramids, and Cones
    4. 10.4Volume of Prisms, Cylinders, Pyramids, and Cones
    5. 10.5Spheres; Cross-Sections and Solids of Revolution
    6. 10.6Effects of Scale on Area and Volume; Density

  11. Unit 11

    Probability and Statistics

    6-8% of exam
    Counting with permutations and combinations, independent and dependent events with and without replacement, conditional probability from two-way tables, probability based on area, and two-variable data displays.
    5 topics
    1. 11.1Sample Spaces, Permutations, and Combinations
    2. 11.2Independent and Dependent Events
    3. 11.3Conditional Probability and Two-Way Tables
    4. 11.4Geometric Probability
    5. 11.5Using Data: Two-Variable Displays and Measurement

The exam, part by part

3 parts, 1 h 55 min in all.

  • Section I, Part A: Multiple Choice

    Questions
    30
    Time
    50 min
    Weight
    50%

    Calculator allowed

    Format details

    Four-option, single-answer items. Roughly a third are set in a real-world context, and several come in sets of 2 or 3 items that share one diagram, graph, or table, the way Pre-AP learning checkpoints and the final exam group items. A graphing calculator or Desmos is available, as on the SAT.

  • Section I, Part B: Student-Produced Response

    Questions
    10
    Time
    20 min
    Weight
    20%

    Calculator allowed

    Format details

    Numeric-entry items only (draw from numeric bank items). Answers follow the SAT entry rules: an integer, decimal, or fraction of at most 5 characters for a positive answer and 6 for a negative one (including the sign); a fraction that does not fit is entered as a decimal, a long decimal is truncated or rounded at the fourth digit, a mixed number becomes an improper fraction, and no symbols such as percent, comma, or dollar signs.

  • Section II: Open Response

    Questions
    2
    Time
    45 min
    Weight
    30%

    Calculator allowed

    Format details

    Task types: proof, constructed-response, performance-task

    One Proof or Constructed Response (about 15 minutes) and one Performance Task (about 30 minutes), scored by the teacher or the AI grader against point-based scoring guidelines.

How the 1 to 5 score is set

The final is reported as a percentage. Multiple choice and student-produced response items are each worth one point with no penalty for guessing; Part A counts for 50 percent, Part B for 20 percent, and the open response for 30 percent. The platform converts the percentage to a 1-5 readiness band (90 percent or higher is a 5, 80-89 a 4, 70-79 a 3, 60-69 a 2, below 60 a 1). This band is a platform convention, not an official score. SAT-tagged items also feed an estimate of SAT Geometry and Trigonometry readiness.

What you bring and get

A graphing calculator or the Desmos graphing calculator on every part, matching the Digital SAT and the Pre-AP graphing-utility requirement. Students receive the reference information the SAT provides (areas and circumferences, the Pythagorean theorem, special right triangles, common volumes, degrees and radians in a circle, and the triangle angle sum). The rest of the course formula sheet is a study aid: the SAT does not print it, so students should know it without the sheet.

Skills the exam scores

  • MODApplications and Modeling

    Build and use geometric and probability models of real situations: choose the relevant measurements, apply the right formula or relationship with correct units, and judge whether the result is reasonable in context.

  • ARGMathematical Argumentation and Proof

    Reason deductively from definitions, postulates, and theorems; write proofs in two-column, paragraph, or flow form; test conjectures and refute them with counterexamples; justify every claim with precise language.

  • REPConnections Among Representations

    Move fluently among diagrams, coordinates, equations, tables, graphs, and words: read a figure into equations, translate a transformation into coordinate notation, or read a probability from a two-way table.

  • TOOLTools, Techniques, and Fluency

    Select and use tools and techniques well: compass-and-straightedge constructions, a graphing calculator or Desmos, mental math, estimation, exact forms such as radicals and multiples of pi, and efficient procedures under time pressure.