Angles and Polygons

These four angle rules are the foundation of all geometric reasoning:

Rule	Statement	Size
Angles at a point	Angles around a full turn	$=360°$
Angles on a straight line	Angles forming a straight line	$=180°$
Vertically opposite	Two lines cross; opposite angles are equal	Equal
Right angle	A perpendicular junction	$=90°$

Worked example: angles at a point are $3x$, $x+20$, and $2x+40$. Find $x$.

$3x+(x+20)+(2x+40)=360⟹6x+60=360⟹x=50°$

Check: $150°+70°+140°=360°$ ✓

Notation (G1): sides of triangle $ABC$ are labelled with lowercase letters matching the opposite vertex — side $a$ is opposite vertex $A$. Angles are written as $\angle ABC$ or $\hat{B}$.

When a transversal crosses two parallel lines (marked $\parallel$), three relationships hold:

Pair type	Description	Size
Alternate angles	Z-shape; on opposite sides of transversal	Equal
Corresponding angles	F-shape; on same side of transversal	Equal
Co-interior (allied) angles	C-shape; between the parallel lines	$=180°$

Worked example: two parallel lines cut by a transversal. One angle is $65°$. Find the alternate angle, the corresponding angle, and the co-interior angle.

• Alternate: $65°$ (Z-pattern)
• Corresponding: $65°$ (F-pattern)
• Co-interior: $180°-65°=115°$ (C-pattern)

Multi-step reasoning: to find angles not in the obvious parallel-line pattern, use a combination of rules (e.g., angles on a straight line, then alternate angles) and show each step with a reason.

Triangle angle sum: angles in any triangle add to $180°$.

Exterior angle theorem: an exterior angle of a triangle equals the sum of the two non-adjacent interior angles.

$\text{Exterior angle}=\alpha +\beta$

Interior angle sum of a polygon with $n$ sides:

$S=(n-2)\times 180°$

Exterior angle sum of any convex polygon $=360°$ (always).

Polygon	$n$	Interior angle sum	Interior angle of regular polygon
Triangle	3	$180°$	$60°$
Quadrilateral	4	$360°$	$90°$
Pentagon	5	$540°$	$108°$
Hexagon	6	$720°$	$120°$
Octagon	8	$1080°$	$135°$

Worked example — each interior angle of a regular polygon is $156°$. How many sides?

Each exterior angle $=180°-156°=24°$. Number of sides $=360°/24°=15$. ✓

Shape	Equal sides	Parallel sides	Equal angles	Diagonals
Square	All 4	2 pairs	All $90°$	Equal, perpendicular bisectors
Rectangle	2 pairs	2 pairs	All $90°$	Equal bisectors
Parallelogram	2 pairs	2 pairs	Opposite equal	Bisect each other
Rhombus	All 4	2 pairs	Opposite equal	Perpendicular bisectors
Trapezium	None	1 pair	—	—
Kite	2 pairs adjacent	0	1 pair opposite	One bisects the other at $90°$

Worked example — in a parallelogram, one angle is $70°$. Find all four angles.

Opposite angles equal: two angles are $70°$. Co-interior angles are supplementary: remaining two are $110°$ each.

$70°+110°+70°+110°=360°$ ✓

Isosceles triangle: two equal sides, base angles equal.

Equilateral triangle: all sides and angles equal ($60°$ each).

Scalene triangle: no equal sides or angles.

Right-angled triangle: one $90°$ angle; hypotenuse is the longest side, opposite the right angle.

Regular polygon: all sides and all angles equal.

Symmetry: a regular $n$-gon has $n$ lines of reflective symmetry and rotational symmetry of order $n$.

Worked example — a regular hexagon. Each interior angle $=\frac{(6-2)\times 180}{6}=\frac{720}{6}=120°$. Lines of symmetry: 6. Rotational order: 6. ✓

(Extra context — a polygon is convex if all interior angles are less than 180°. Reflex interior angles produce a concave (non-convex) polygon. The interior angle sum formula works for both, provided the angles are measured correctly.)

1. Alternate and co-interior confusion

Alternate angles are equal (Z-shape). Co-interior angles sum to 180° (C-shape). Confusing the two — writing equal co-interior angles or summing alternate angles — is a common error.

2. Polygon angle sum — using $n$ instead of $(n-2)$

For a pentagon: $(5-2)\times 180=540°$, not $5\times 180=900°$. The formula subtracts 2 because the triangle fan uses $n-2$ triangles.

3. Regular polygon — dividing the sum by $n-2$ instead of $n$

Interior angle sum of a hexagon is $720°$. Each interior angle $=720/6=120°$ (divide by $n$, not $n-2$).

4. Not stating angle reasons

Exam mark schemes require reasons: write "alternate angles" or "angles on a straight line" alongside each step. A correct numerical answer without reasons loses marks in angle-proof questions.

Mistake	Correction
"Vertically opposite angles are supplementary"	Vertically opposite angles are equal
"Co-interior angles are equal"	Co-interior angles sum to 180°
"Interior angle sum of octagon: $8\times 180=1440°$"	$(8-2)\times 180=1080°$