Properties and Constructions

Basic angle facts:

Fact	Rule
Angles on a straight line	Sum to $180°$
Angles at a point	Sum to $360°$
Vertically opposite angles	Equal

Angles on parallel lines — always state which rule you are using in proofs:

$\text{Alternate angles (Z-angles)} = \text{equal}$ $\text{Corresponding angles (F-angles)} = \text{equal}$ $\text{Co-interior angles (C-angles)} = 180°$

Proving lines are parallel: if you can show that alternate or corresponding angles are equal (or co-interior sum to 180°), the lines must be parallel.

Setting up equations:

Find the value of $x$ if two alternate angles are $(3x + 10)°$ and $(5x - 20)°$.

$3x + 10 = 5x - 20 \implies 30 = 2x \implies x = 15$

Bearings (AQA includes these here): measured clockwise from north, always written as three figures (e.g. 045°, 270°). Use alternate/co-interior angle rules to solve bearing problems with parallel north lines.

Angle sums:

$\text{Sum of interior angles of an } n\text{-gon} = (n - 2) \times 180°$

$\text{Sum of exterior angles of any polygon} = 360°$

For a regular polygon with $n$ sides:

$\text{Each interior angle} = \frac{(n-2) \times 180°}{n}$

$\text{Each exterior angle} = \frac{360°}{n}$

Interior angle + exterior angle = 180° (angles on a straight line).

Finding the number of sides given an exterior angle: $n = 360° \div \text{exterior angle}$.

Properties of special quadrilaterals:

Shape	Key properties
Square	4 equal sides; 4 right angles; 4 lines of symmetry; rotation order 4
Rectangle	Opposite sides equal; 4 right angles; 2 lines of symmetry; diagonals bisect each other
Parallelogram	Opposite sides equal and parallel; opposite angles equal; diagonals bisect each other
Rhombus	4 equal sides; opposite angles equal; diagonals bisect each other at 90°; 2 lines of symmetry
Trapezium	One pair of parallel sides; co-interior angles between parallel sides sum to 180°
Kite	Two pairs of adjacent equal sides; one pair of equal angles; one diagonal bisects the other at 90°

Triangle types and properties:

• Equilateral: all sides equal, all angles 60°
• Isosceles: two equal sides, base angles equal
• Scalene: no equal sides or angles
• Right-angled: one angle of 90°

Congruence means identical in shape and size. Two triangles are congruent if any one of the following criteria is satisfied:

Criterion	Meaning
SSS	All three sides equal
SAS	Two sides and the included angle equal
ASA (or AAS)	Two angles and a corresponding side equal
RHS	Right angle, hypotenuse and one other side equal

Note: AAA is not a congruence criterion — it proves similarity, not congruence.

Writing a congruence proof:

1. State the three pieces of matching information and give reasons for each.
2. State the congruence criterion used.
3. Conclude: "Therefore triangle $ABC \equiv$ triangle $DEF$ (SSS/SAS/ASA/RHS)."

Similarity means the same shape but different size. Two triangles are similar if:

• All three pairs of angles are equal (AA is sufficient, since the third follows)
• All three pairs of sides are in the same ratio

Scale factor: if linear scale factor is $k$:

• Length scale factor: $k$
• Area scale factor: $k^2$
• Volume scale factor: $k^3$

Example: Two similar triangles have areas 25 cm² and 100 cm². Find the ratio of their corresponding sides.

Area ratio $= 100 : 25 = 4 : 1$, so $k^2 = 4 \implies k = 2$. Sides are in ratio $2 : 1$.

All constructions must be done with a ruler and compasses only — no protractor. Leave all construction arcs visible for the examiner.

Standard constructions:

1. Perpendicular bisector of line segment $AB$:

• Open compasses to more than half of $AB$
• Draw arcs above and below from both $A$ and $B$
• Join the two intersection points — this line is the perpendicular bisector
• The locus of points equidistant from $A$ and $B$

2. Angle bisector of angle $ABC$:

• Draw arcs from $B$ on both arms to get points $P$ and $Q$
• Draw equal arcs from $P$ and $Q$ to intersect at $R$
• Join $BR$ — this is the angle bisector
• The locus of points equidistant from lines $BA$ and $BC$

3. Perpendicular from point $P$ to line $l$:

• Open compasses from $P$ to cross line $l$ at two points $A$ and $B$
• Construct the perpendicular bisector of $AB$

4. Perpendicular at point $P$ on line $l$:

• Draw arcs from $P$ to get equal points $A$ and $B$ on the line
• Construct the perpendicular bisector of $AB$

Loci — common types:

Description	Locus
Fixed distance $r$ from point $P$	Circle, centre $P$, radius $r$
Equidistant from two points $A$ and $B$	Perpendicular bisector of $AB$
Equidistant from two lines	Angle bisector
Fixed distance from a line	Two parallel lines (one each side)

Combining loci: shade or outline the region satisfying all conditions simultaneously.

You must know all six circle theorems for AQA Higher, and be able to state the theorem when using it as a reason.

Theorem 1 — Angle at the centre: The angle subtended at the centre is twice the angle subtended at the circumference by the same arc.

$\angle AOB = 2 \times \angle ACB \quad (\text{where } O \text{ is the centre})$

Theorem 2 — Angles in the same segment: Angles subtended by the same arc at the circumference are equal.

$\angle ACB = \angle ADB \quad (\text{both on major arc } AB)$

Theorem 3 — Angle in a semicircle: The angle in a semicircle (subtended by a diameter) is 90°.

$AB \text{ is a diameter} \implies \angle ACB = 90°$

Theorem 4 — Cyclic quadrilateral: Opposite angles of a cyclic quadrilateral sum to 180°.

$\angle A + \angle C = 180°, \quad \angle B + \angle D = 180°$

Theorem 5 — Tangent and radius: A tangent to a circle is perpendicular to the radius at the point of tangency.

$OT \perp \text{tangent} \quad (\text{where } O \text{ is centre, } T \text{ is point of tangency})$

Theorem 6 — Alternate segment theorem: The angle between a tangent and a chord equals the angle in the alternate segment.

$\angle \text{(tangent-chord)} = \angle \text{(inscribed angle in alternate segment)}$

Other circle facts:

• Two tangents from the same external point are equal in length
• A radius that bisects a chord is perpendicular to it (and vice versa)
• The perpendicular from the centre to a chord bisects the chord

Proof strategy for circle theorem questions:

1. Mark all known angles on the diagram
2. Identify which theorem to apply at each step
3. State the theorem as a reason (e.g. "angle at centre = twice angle at circumference")
4. Work systematically toward the required angle

Properties of common 3D solids:

Solid	Faces	Edges	Vertices
Cube	6	12	8
Cuboid	6	12	8
Triangular prism	5	9	6
Square-based pyramid	5	8	5
Cylinder	3 (2 circles + 1 curved)	2	0
Cone	2 (1 circle + 1 curved)	1	1
Sphere	1 curved	0	0

Euler's formula (for polyhedra): $F + V = E + 2$

Plans and elevations:

Three standard views are used to represent a 3D shape on a 2D page:

View	Description
Plan	Looking directly down (bird's-eye view)
Front elevation	Looking directly from the front
Side elevation	Looking directly from one side

Reading plans and elevations:

• Solid lines show visible edges; dashed lines show hidden edges
• All three views are drawn to the same scale
• From three views, reconstruct the 3D shape

Cross-sections of prisms: a prism has a uniform cross-section throughout its length. The cross-section is always the same shape wherever you slice perpendicular to the length.