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What you will learn in Euclidean Geometry & Proof
1. Points, Lines and Angles
The vocabulary, and the habit that makes the rest of the module possible: never state an angle without stating why.
2. Parallel Lines and a Transversal
Which angle equalities need the lines to be parallel, and which were never about parallelism at all.
3. Congruent Triangles
Five sets of measurements that force one triangle, and one that does not — shown as two triangles rather than described.
4. Sides, Angles and the Triangle Inequality
Three lengths either close or they do not, and once they close the biggest angle faces the longest side.
5. Similar Figures and Scale Factors
Length goes as k, area as k squared, volume as k cubed — and assuming otherwise is the subject's most persistent error.
6. Pythagoras, Twice
Two independent proofs, because one proof shows a statement is true and two show why it could not have been otherwise.
7. Quadrilaterals and Their Hierarchy
A square is a rectangle that satisfies one more condition. The names are conditions, not pictures.
8. Circles and Chords
Two radii and a chord make an isosceles triangle, and the whole chapter falls out of that.
9. Angles in a Circle
Quantities that refuse to move however hard you drag — and what to make of that.
10. Tangents
A tangent meets the radius at a right angle, and two tangents from a point are equal — until the point arrives at the circle.
11. Constructions, and Why They Work
A construction is checked on method. Getting the right picture by measuring is a different thing, and the checker can tell.
12. The Shape of a Proof
Statements, reasons, and an order in which each line follows from the ones above it.
13. Loci
State a condition and see what satisfies it. The shape is the consequence, not the starting point.