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Grade 10 – Sophomore Year

Geometry

Where mathematics stops calculating and starts proving: shape, space, measurement and the logic that establishes them.
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About Course

Geometry shifts mathematics from calculation to reasoning. You will study shape,
space and measurement, but the deeper subject of the course is proof — learning how a
mathematical claim is established beyond doubt rather than merely checked.

This is the only place in school mathematics where you learn what a valid argument actually is: what
counts as a reason, why a converse does not follow from a conditional, and how a chain of justified
steps produces certainty. Those habits transfer far beyond mathematics.

What you will study

  • Unit 1 — Foundations: undefined terms, segments and angles, angle pairs, and
    compass-and-straightedge constructions.
  • Unit 2 — Reasoning and Proof: conditional statements, inductive and deductive
    reasoning, two-column proofs, and parallel lines.
  • Unit 3 — Triangles: classification, angle relationships, the five congruence
    criteria, isosceles triangles and inequalities.
  • Unit 4 — Similarity and Right Triangles: ratio and proportion, similarity
    criteria, the Pythagorean Theorem, and right triangle trigonometry.
  • Unit 5 — Polygons and Circles: angle sums, special quadrilaterals, and circle
    theorems involving chords, tangents, arcs and inscribed angles.
  • Unit 6 — Measurement and Transformations: area and volume, coordinate geometry,
    and the transformations that define congruence and similarity.

How the course works

Most of your time is spent on teaching lessons containing full worked examples and complete proofs.
Each unit includes two interactive activities you complete on screen and one unit quiz marked
automatically. A mid-course assessment follows Unit 3 and a final assessment closes the course.

What you will be able to do

By the end you will be able to construct a valid argument from stated assumptions, measure what you
cannot reach, and recognise when a claim has been established rather than merely asserted.

Show More

What Will You Learn?

  • Construct valid logical arguments and formal proofs
  • Apply congruence and similarity criteria to solve problems
  • Use the Pythagorean Theorem and trigonometry for indirect measurement
  • Calculate area, surface area and volume, and apply scale factor correctly
  • Use coordinate methods and transformations to analyse figures

Requirements

  • Algebra I or equivalent. Confidence with equations, square roots and the coordinate plane.

Audience

  • Students who have completed Algebra I, particularly those considering pathways in engineering, architecture, design, trades or computer science.

Course Content

Getting Started
<p>How this course works, what you will learn, and how you are assessed.</p>

  • Course Overview and Learning Objectives

Unit 1: Foundations of Geometry
<h2>Unit Introduction</h2> <p>Geometry begins with a problem that sounds like a technicality and is not: <strong>you cannot define everything.</strong></p> <p>Every definition uses words, and those words need defining, which uses more words. Eventually you either run in circles or you stop. Geometry stops deliberately, at three terms it refuses to define — point, line and plane — and builds everything else from them.</p> <p>That decision is what makes geometry different from the mathematics you have done before. <strong>Algebra asks what the answer is. Geometry asks how you know</strong>, and the answer always traces back to something agreed at the start.</p> <h2>Unit Learning Objectives</h2> <p>By the end of Unit 1 you will be able to: - Use the undefined terms and standard geometric notation correctly - Measure and calculate with segments and angles - Identify angle pair relationships and use them to find unknown measures - Perform basic compass-and-straightedge constructions and explain why they work</p>

Unit 2: Reasoning and Proof
<h2>Unit Introduction</h2> <p>This is the unit that makes geometry different from every mathematics course you have taken.</p> <p>Elsewhere you calculate. Here you <strong>establish</strong> — you show that something must be true, given what has already been agreed. The skill is not arithmetic. It is constructing a chain of reasoning where every link is justified.</p> <p><strong>Students find this the hardest part of the course and the most useful afterwards</strong>, because it is the only place in school mathematics where you learn what a valid argument actually is.</p> <h2>Unit Learning Objectives</h2> <p>By the end of Unit 2 you will be able to: - Write and analyse conditional statements and their variations - Distinguish inductive from deductive reasoning - Write two-column proofs using definitions, postulates and theorems - Prove and apply relationships created by parallel lines and a transversal</p>

Unit 3: Triangles
<h2>Unit Introduction</h2> <p>The triangle is the most important figure in geometry, for a reason that is worth stating plainly: <strong>it is rigid.</strong> Fix the three side lengths and the shape is determined — there is exactly one triangle with those sides. No other polygon behaves this way. A quadrilateral with fixed sides can flex into infinitely many shapes.</p> <p>That rigidity is why triangles appear in bridges, roof trusses, bicycle frames and cranes, and it is why triangle congruence criteria exist at all.</p> <h2>Unit Learning Objectives</h2> <p>By the end of Unit 3 you will be able to: - Classify triangles by sides and angles - Apply the Triangle Sum and Exterior Angle Theorems - Prove triangles congruent using SSS, SAS, ASA, AAS and HL - Apply isosceles triangle theorems and the triangle inequality</p>

Unit 4: Similarity and Right Triangles
<h2>Unit Introduction</h2> <p>Congruence asks whether two figures are identical. <strong>Similarity asks whether they are the same shape at different sizes</strong> — and that turns out to be the more useful question, because it is how maps, scale models, photographs and shadows all work.</p> <p>This unit also reaches the Pythagorean Theorem and the beginning of trigonometry, which is where geometry becomes a tool for measuring things you cannot reach.</p> <h2>Unit Learning Objectives</h2> <p>By the end of Unit 4 you will be able to: - Work with ratios and proportions, including scale factor - Prove triangles similar using AA, SSS and SAS similarity - Apply the Pythagorean Theorem and its converse - Use right triangle trigonometry to find unknown sides and angles</p>

Unit 5: Polygons, Quadrilaterals and Circles
<h2>Unit Introduction</h2> <p>This unit extends what you have established about triangles to figures with more sides, and then to the circle — which has no sides at all and behaves differently from everything before it.</p> <p>The quadrilateral section is largely about <strong>classification and hierarchy</strong>: which shapes are special cases of which others, and what follows from being in a particular category. The circle section introduces a new set of relationships that are worth learning properly, because they underpin a great deal of later mathematics.</p> <h2>Unit Learning Objectives</h2> <p>By the end of Unit 5 you will be able to: - Calculate interior and exterior angles of polygons - Identify special quadrilaterals and apply their properties - Apply relationships involving chords, tangents, arcs and inscribed angles - Solve problems using circle theorems</p>

Unit 6: Measurement, Coordinates and Transformations
<h2>Unit Introduction</h2> <p>The final unit measures the figures you have been studying and then moves them.</p> <p><strong>Coordinate geometry</strong> merges the algebra of Algebra I with the reasoning of this course — every geometric claim becomes an algebraic calculation, and proofs that were difficult synthetically become routine. <strong>Transformations</strong> formalise what it means for two figures to be congruent or similar, and close the course by explaining what congruence actually was all along.</p> <h2>Unit Learning Objectives</h2> <p>By the end of Unit 6 you will be able to: - Calculate area and perimeter of plane figures - Calculate surface area and volume of solids - Apply coordinate methods to prove geometric statements - Identify and perform transformations, and describe symmetry</p>

Course Completion
<p>Final assessment and full-course review.</p>

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Rapid Grad Academy

Rapid Grad Academy

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90 Students
35 Courses
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$75.00
  • Grade 10 – Sophomore Year
  • 1
  • September 26, 2026
  • Certificate of completion

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