Geometry

Deductive Reasoning and Triangles

Geometry justifies conclusions through logical steps. Five axioms are accepted without proof: exactly one line passes through two distinct points; distances are non-negative. If C lies between A and B, $|AB| = |AC| + |CB|$; adjacent angles forming a straight angle total $180^\circ$; triangles are congruent by SAS, ASA, or SSS; and exactly one line parallel to a given line passes through an external point.

Key angle and triangle rules:

• Vertically opposite angles are equal.
• Alternate and corresponding angles on parallel lines are equal; their converses prove lines are parallel.
• Angles in a triangle sum to $180^\circ$; an exterior angle equals the sum of opposite interior angles.
• In an isosceles triangle, angles opposite equal sides are equal; the converse also holds.
• Triangles are similar if angles are equal, which implies corresponding sides are proportional.
• In a right-angled triangle, $c$ is the hypotenuse, opposite the right angle: $c^2 = a^2 + b^2$.

Polygons, Circles, and Constructions

• Parallelograms have equal opposite sides and angles; diagonals bisect each other and divide the shape into congruent triangles.
• If three parallel lines cut equal segments on one transversal, they cut equal segments on any transversal.
• The angle at the centre of a circle is twice the angle at the circumference on the same arc.
• Angles on the same arc are equal; the angle in a semicircle is $90^\circ$.
• Conversely, a chord standing opposite a right angle at the circumference is a diameter.
• In a cyclic quadrilateral, opposite angles sum to $180^\circ$; the converse also holds.
• A tangent touches a circle at one point; lines meeting at one point are concurrent.
• Use the tools permitted for each construction, including a protractor where appropriate; constructions 1–2 require only compass and straightedge.
• Higher Level adds constructions 3 (perpendicular through external point) and 7 (dividing a segment into equal parts without measuring).

2D Measurement, Perimeter, and Area

Always convert all measurements to matching units before calculating ($1\text{ m} = 100\text{ cm}$, $1\text{ m}^2 = 10{,}000\text{ cm}^2$). On a scaled diagram of $1:50$, $1\text{ cm}$ represents $50\text{ cm}$ in reality.

• Rectangle: $\text{Area} = lw$; $\text{Perimeter} = 2(l + w)$.
• Triangle: $\text{Area} = \frac{1}{2}bh$, where $h$ is the perpendicular height.
• Disc: $\text{Area} = \pi r^2$; circumference $= 2\pi r = \pi d$.
• Compound figures: split the shape into rectangles, triangles, and discs, then add the separate areas, or subtract cut-out sections.
• Perimeter is the total distance around the outer boundary only; do not include internal dividing lines.

3D Solids, Surface Area, and Nets

Volume measures the space a solid occupies; capacity measures how much a container holds ($1\text{ cm}^3 = 1\text{ ml}$, $1{,}000\text{ cm}^3 = 1\text{ litre}$).

• Rectangular solid (cuboid): $V = lwh$; $\text{Surface Area} = 2(lw + lh + wh)$.
• Cylinder: $V = \pi r^2 h$; curved surface area $= 2\pi rh$; closed cylinder total surface area $= 2\pi rh + 2\pi r^2$.
• Triangular-based prism: $V = \text{base area} \times \text{length} = Ah$; surface area $= 2A + PL$, where $P$ is the triangle's perimeter and $L$ is length.
• Sphere: $V = \frac{4}{3}\pi r^3$; surface area $= 4\pi r^2$.
• For composite solids, add or subtract volumes, but for surface area count only exposed external faces.
• Nets show unfolded shapes: a cuboid has six rectangles; a cylinder net has two discs and a $2\pi r \times h$ rectangle.

Coordinate Geometry and Transformations

Coordinate formulas link geometry with algebra:

• Distance: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$; midpoint: $M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$.
• Slope $m = \frac{y_2 - y_1}{x_2 - x_1}$. On real-world graphs, slope represents the rate of change and the $y$-intercept is the starting value.
• Parallel non-vertical lines have equal slopes ($m_1 = m_2$). Perpendicular lines with defined slopes satisfy $m_1 \times m_2 = -1$.
• Line equations: $y = mx + c$, $y - y_1 = m(x - x_1)$, and general form $ax + by + c = 0$. Solve simultaneous equations to find points of intersection.
• Transformations preserve lengths and angles: translation (slide), axial symmetry (reflection in a line), central symmetry (half-turn through a point), and rotation (turn about a centre).
• An axis of symmetry folds a shape into matching reflected halves.

Key terms

Axiom

A foundational statement accepted as true without proof, used as a starting rule for deduction.

Theorem

A mathematical statement proved to be true by a chain of logical deduction from axioms and established results.

Corollary

A mathematical result that follows directly and easily from an established theorem.

Converse

A statement formed by swapping the 'if' and 'then' parts of a proposition.

Congruent Triangles

Triangles with equal corresponding sides and equal corresponding angles.

Similar Triangles

Triangles with equal corresponding angles and proportional corresponding sides.

Cyclic Quadrilateral

A four-sided polygon whose four vertices all lie on a circle.

Tangent

A straight line that touches a circle at exactly one point, called the point of contact.