Geometry

What this covers

• Writing two-column and paragraph proofs for triangle congruence using SSS, SAS, ASA, AAS and HL, then extending to CPCTC conclusions
• Angle relationships from parallel lines cut by a transversal, triangle and polygon angle sums, and exterior angle arguments
• Similarity: AA criterion, proportional side lengths, geometric mean relationships in right triangles, and how a scale factor k affects perimeter, area and volume
• Right-triangle trigonometry — sine, cosine, tangent ratios, inverse trig to find angles, and special right triangles (30-60-90, 45-45-90)
• Circle theorems: central versus inscribed angles, tangent-radius perpendicularity, chord and secant segment relationships, arc length and sector area
• Coordinate geometry proofs and transformations — distance and midpoint formulas, slope tests for parallel/perpendicular, reflections, rotations, translations and dilations

Where learners get stuck

Treating SSA as a valid congruence shortcut

Students memorise the list of letter combinations rather than the reasoning behind them, and SSA looks structurally identical to SAS. Sketching the two triangles that share the same SSA data makes the ambiguity visible, and clarifies why HL is a legitimate special case.

Reading measurements off the picture instead of the given information

Diagrams are drawn to look reasonable, so a nearly-right angle reads as 90° and two nearly-equal segments read as congruent. Proof requires that every claim trace back to a given, a definition, or a previously proved statement — a habit that takes deliberate practice to build.

Applying the scale factor directly to area and volume

After weeks of setting up side-length proportions, it feels consistent to say that doubling the sides doubles the area. Working through a concrete rectangle or cube, then generalising to k² and k³, is usually what makes the distinction stick.

What a session looks like

Sessions run as spoken conversation. You describe the figure you are looking at — or Evelyn Tutor describes one for you — and then you talk through the reasoning step by step: what is given, what the goal is, and which theorem links them. For proofs, you say each statement and its reason aloud before writing it down, which surfaces gaps in the chain quickly. For computation-heavy work like trig ratios or volume, you set up the equation verbally and Evelyn Tutor checks the setup before you grind through the arithmetic. Having paper, a pencil and a ruler nearby matters here more than in most subjects; you will be sketching constantly.

Helpful to know first

• Solving linear equations and simple proportions, since most geometry problems end in an algebraic step
• Comfort with square roots and simplifying radicals, which appear throughout distance, Pythagorean and special-triangle work
• The Pythagorean theorem and basic area formulas from middle school mathematics
• Plotting points and reading slope on the coordinate plane

Questions

How can geometry be taught over voice when it is such a visual subject?

You keep the diagram in front of you on paper or on screen and describe what you see; Evelyn Tutor asks targeted questions about labels, markings and given information. Much of geometry difficulty is verbal reasoning — stating why a step follows — and speaking that reasoning out loud tends to expose errors faster than writing does.

My child can do the calculations but freezes on proofs. What helps?

Proof difficulty is usually about not knowing where to start rather than not knowing the theorems. Sessions focus on the working-backwards habit: name the conclusion, name what would be sufficient to reach it, and repeat until you land on something given. Congruence proofs are the usual entry point because the toolkit is small and finite.

Does this cover the geometry on state end-of-course exams?

Sessions cover the standard grade 9–10 geometry content — congruence, similarity, circles, right-triangle trigonometry, coordinate geometry, transformations and solid measurement — which is what those exams draw from. Evelyn Tutor has no affiliation with any testing body, so bring your own syllabus or released questions and work through them directly.

Is geometry usually taken before or after Algebra 2?

Both sequences are common. Geometry uses Algebra 1 skills — solving equations, proportions, radicals — but very little from Algebra 2, so either order works. If your school interleaves them, the coordinate geometry and trigonometry units are where the algebra overlap is heaviest.