Geometric Conventions and Notation

Geometry has its own vocabulary. Using the right word is worth marks, because examiners need to know exactly which object you mean.

Word	Meaning
Point	A position with no size. Labelled with a capital letter, e.g. $A$.
Line	Straight and infinitely long in both directions.
Line segment	The part of a line between two points, e.g. $AB$. This is what you actually draw.
Ray	Starts at a point and continues forever in one direction.
Plane	A flat surface extending forever — the page you are drawing on.
Vertex	A corner, where two edges meet. Plural: vertices.
Edge	A line segment where two faces of a solid meet.
Face	A flat surface of a solid.
Parallel	Always the same distance apart; never meet. Written $AB \parallel CD$.
Perpendicular	Meeting at $90^\circ$. Written $AB \perp CD$.

Diagram conventions:
• Arrows ► on two lines mean they are parallel. Double arrows ►► mark a second parallel pair.
• Dashes across sides mean those sides are equal. Double dashes mark a second equal pair.
• A small square in a corner means a right angle.
• Arcs inside angles mean those angles are equal.

There is a standard convention that every exam board uses, and understanding it makes questions much easier to read.

The standard convention

Vertices get CAPITAL letters: $A$, $B$, $C$.
Sides get the lower-case letter of the opposite vertex: $a$, $b$, $c$.

Notation	Means
$AB$	The line segment from $A$ to $B$, or its length.
$\angle ABC$ or $A\hat{B}C$	The angle at $B$, formed by $BA$ and $BC$.
$\triangle ABC$	The triangle with vertices $A$, $B$ and $C$.
$AB \parallel CD$	$AB$ is parallel to $CD$.
$AB \perp CD$	$AB$ is perpendicular to $CD$.

The middle letter is the angle. In $\angle ABC$ the vertex is $B$ — the letter in the middle. $\angle BAC$ is a completely different angle, the one at $A$.

Worked Example 1 — Reading the notation

In triangle $PQR$, $\angle QPR = 55^\circ$ and $\angle PQR = 70^\circ$. Find $\angle PRQ$.

①$\angle QPR$ is the angle at $P$ (middle letter), so $P = 55^\circ$.

②$\angle PQR$ is the angle at $Q$, so $Q = 70^\circ$.

③$\angle PRQ$ is the angle at $R$.

④Angle sum of a triangle: $R = 180 - 55 - 70 = 55^\circ$.

Angle type	Size
Acute	Less than $90^\circ$
Right angle	Exactly $90^\circ$
Obtuse	Between $90^\circ$ and $180^\circ$
Straight angle	Exactly $180^\circ$
Reflex	Between $180^\circ$ and $360^\circ$

Triangle	Sides	Angles
Equilateral	All three equal	All $60^\circ$
Isosceles	Two equal	Two equal (the base angles)
Scalene	All different	All different
Right-angled	—	One angle is $90^\circ$

A triangle can have two labels. A right-angled isosceles triangle has angles $90^\circ$, $45^\circ$, $45^\circ$ — it is both right-angled and isosceles.

A polygon is a closed 2D shape made only of straight sides.

Sides	Name	Sides	Name
3	Triangle	8	Octagon
4	Quadrilateral	9	Nonagon
5	Pentagon	10	Decagon
6	Hexagon	11	Hendecagon
7	Heptagon	12	Dodecagon

Regular means all sides equal AND all angles equal. Both conditions are needed.
A rhombus has all sides equal but is not regular, because its angles are not all equal.
A rectangle has all angles equal but is not regular, because its sides are not all equal.
A square is both — so a square is a regular quadrilateral.

Convex polygons have every interior angle less than $180^\circ$.
Concave (or re-entrant) polygons have at least one reflex interior angle — they look "dented".

Line symmetry (reflection symmetry): a mirror line can be drawn so that one half maps exactly onto the other.
Rotational symmetry: the shape looks identical after turning through less than a full turn. The order is the number of positions in which it looks the same during one complete turn.

Shape	Lines of symmetry	Rotational order
Equilateral triangle	$3$	$3$
Isosceles triangle	$1$	$1$
Scalene triangle	$0$	$1$
Square	$4$	$4$
Rectangle	$2$	$2$
Rhombus	$2$	$2$
Parallelogram	$0$	$2$
Kite	$1$	$1$
Regular $n$-gon	$n$	$n$
Circle	Infinitely many	Infinite

Rotational order is never $0$. Every shape returns to itself after a full $360^\circ$ turn, so the minimum order is $1$ — which means "no rotational symmetry".

Worked Example 2 — Symmetry of a parallelogram

Explain why a parallelogram has rotational symmetry of order $2$ but no lines of symmetry.

①Rotating $180^\circ$ about the centre sends each vertex to the opposite one, and the shape looks identical.

②Together with the $360^\circ$ position, that gives order $2$.

③Folding along either diagonal does not match the halves, because the angles at the ends of a diagonal are different.

④Folding along a line through the midpoints of opposite sides fails too, because the shape leans.

So there are $0$ lines of symmetry.

Exam questions often give you words and no picture. Drawing your own diagram is nearly always the fastest route to the answer.

Read the whole description before drawing anything.
Sketch the main shape roughly — it does not need to be accurate unless the question says so.
Add every label, in the order the letters are given.
Mark equal sides with dashes, equal angles with arcs and right angles with squares.
Write on every measurement you are told.
Check your diagram against the description sentence by sentence.

Worked Example 3 — Building a diagram

"$ABCD$ is a trapezium with $AB$ parallel to $DC$. $AD = BC$. Angle $ADC = 68^\circ$." Draw and label the diagram, then find angle $DAB$.

①Draw $DC$ as the longer bottom side and $AB$ shorter and parallel above it, with $A$ above $D$ and $B$ above $C$.

②Mark $AD$ and $BC$ with single dashes to show they are equal — this is an isosceles trapezium.

③Write $68^\circ$ at $D$.

④$AB \parallel DC$, so angles $DAB$ and $ADC$ are co-interior and add to $180^\circ$.

⑤Angle $DAB = 180 - 68 = 112^\circ$.

Letters go round the shape in order. In quadrilateral $ABCD$ the sides are $AB$, $BC$, $CD$ and $DA$; $AC$ and $BD$ are the diagonals. Drawing the letters out of order will make everything else wrong.

Angle notation

$\angle ABC$ is the angle at $B$ — the middle letter.

Triangle labels

Capital for a vertex, matching lower case for the opposite side.

Diagram marks

Arrows = parallel; dashes = equal sides; arcs = equal angles; square = $90^\circ$.

Regular polygon

All sides equal AND all angles equal — both conditions.

Angle names

Acute $\lt 90^\circ$, right $=90^\circ$, obtuse $90$–$180^\circ$, reflex $180$–$360^\circ$.

Rotational order

Counts positions in a full turn; minimum value is $1$.

Letters in order

$ABCD$ goes round the shape; $AC$ and $BD$ are diagonals.

Sketch it

Always draw your own diagram if the question gives only words.

Question 1

In triangle $XYZ$, angle $XYZ = 48^\circ$ and angle $YZX = 63^\circ$. Find angle $ZXY$, and state which vertex each angle is at.

▶ Show solution

$\angle XYZ$ is at $Y$ (middle letter) $= 48^\circ$.

$\angle YZX$ is at $Z = 63^\circ$.

$\angle ZXY$ is at $X$.

Angle sum of a triangle: $180 - 48 - 63 = 69^\circ$.

Question 2

Classify each angle: (a) $89^\circ$ (b) $90^\circ$ (c) $174^\circ$ (d) $271^\circ$.

▶ Show solution

(a) Acute (less than $90^\circ$)

(b) A right angle

(c) Obtuse (between $90^\circ$ and $180^\circ$)

(d) Reflex (between $180^\circ$ and $360^\circ$)

Question 3

State the number of lines of symmetry and the order of rotational symmetry of (a) a regular hexagon, (b) a kite, (c) a parallelogram.

▶ Show solution

(a) Regular hexagon: $6$ lines, order $6$.

(b) Kite: $1$ line (along the long diagonal), order $1$.

(c) Parallelogram: $0$ lines, order $2$.

Question 4

Explain why a rhombus is not a regular polygon, even though all four of its sides are equal.

▶ Show solution

A regular polygon needs both all sides equal and all angles equal.

A rhombus has four equal sides, but its angles come in two different pairs (two acute and two obtuse) unless it happens to be a square.

Because the angles are not all equal, it is not regular.

Question 5

A shape has $12$ sides. (a) Name it. (b) If it is regular, state its number of lines of symmetry and rotational order.

▶ Show solution

(a) A dodecagon.

(b) A regular $n$-gon has $n$ lines of symmetry and rotational order $n$, so $12$ lines and order $12$.

Question 6

$PQRS$ is a quadrilateral. Write down (a) the four sides, (b) the two diagonals.

▶ Show solution

(a) Sides: $PQ$, $QR$, $RS$, $SP$ — the letters taken in order round the shape.

(b) Diagonals: $PR$ and $QS$ — joining opposite vertices.

Question 7

Draw and label a diagram for: "Triangle $LMN$ has $LM = LN$, angle $MLN = 36^\circ$, and $MN$ is horizontal." Then find angle $LMN$.

▶ Show solution

Draw $MN$ horizontally with $L$ above the midpoint. Mark $LM$ and $LN$ with single dashes (equal), and write $36^\circ$ at $L$.

$LM = LN$ so the triangle is isosceles and the base angles $LMN$ and $LNM$ are equal.

$180 - 36 = 144^\circ$ shared between them.

Angle $LMN = 144 \div 2 = 72^\circ$.

Question 8

A triangle has one angle of $90^\circ$ and one line of symmetry. Find all three of its angles and name the triangle fully.

▶ Show solution

One line of symmetry means it is isosceles, so two angles are equal.

The $90^\circ$ angle cannot be one of the equal pair (two right angles would already total $180^\circ$).

Remaining: $180 - 90 = 90^\circ$ split equally, so $45^\circ$ each.

Angles: $90^\circ$, $45^\circ$, $45^\circ$ — a right-angled isosceles triangle.

Question 9

Write each statement in symbols: (a) $AB$ is parallel to $CD$. (b) $PQ$ is perpendicular to $QR$. (c) The angle at $Y$ in triangle $XYZ$ is $50^\circ$.

▶ Show solution

(a) $AB \parallel CD$

(b) $PQ \perp QR$

(c) $\angle XYZ = 50^\circ$ (the vertex $Y$ must be the middle letter)

Question 10

A quadrilateral has exactly $2$ lines of symmetry and rotational symmetry of order $2$. Its diagonals cross at right angles.

(a) Name the shape. (b) Explain how you know it is not a rectangle. (c) What extra condition would make it a square?

▶ Show solution

(a) A rhombus. Its two lines of symmetry are its diagonals, and it has rotational order $2$.

(b) A rectangle also has $2$ lines of symmetry and order $2$, but its diagonals do not cross at right angles (unless it is a square). The perpendicular diagonals rule the rectangle out.

(c) If the angles were also all equal ($90^\circ$ each) — equivalently if the diagonals were also equal in length — the rhombus would be a square, with $4$ lines of symmetry and order $4$.