Symmetry Patterns

Symmetry is one of the more abstract pattern categories in UCAT AR. While most candidates intuitively understand that a butterfly has symmetry, applying symmetry concepts under time pressure to collections of abstract shapes requires deliberate practice. This lesson covers both line symmetry (reflective) and rotational symmetry, and shows how each appears in AR questions.

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Two Types of Symmetry

Line Symmetry (Reflective Symmetry)

A shape or arrangement has line symmetry if you can draw a line through it such that one side is a mirror image of the other. This line is called the axis of symmetry or line of symmetry.

Shape	Lines of symmetry
Circle	Infinite
Square	4 (two diagonals, one horizontal, one vertical)
Rectangle	2 (one horizontal, one vertical)
Equilateral triangle	3
Isosceles triangle	1
Scalene triangle	0
Regular pentagon	5
Regular hexagon	6
Parallelogram	0
Arrow (symmetric)	1 (along its length)

Rotational Symmetry

A shape has rotational symmetry if you can rotate it by some angle less than 360° and it looks identical to its original orientation. The order of rotational symmetry is the number of times it looks the same during a full 360° rotation.

Shape	Order of rotational symmetry
Circle	Infinite
Square	4 (every 90°)
Rectangle	2 (every 180°)
Equilateral triangle	3 (every 120°)
Regular pentagon	5 (every 72°)
Regular hexagon	6 (every 60°)
Parallelogram	2 (every 180°)
Scalene triangle	1 (only at 360° — effectively no rotational symmetry)

Key distinction: A shape can have line symmetry without rotational symmetry (e.g., an isosceles triangle), and rotational symmetry without line symmetry (e.g., a parallelogram). They are independent properties.

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Category 1: Individual Shape Symmetry Rules

The rule concerns the symmetry properties of individual shapes in each box.

Common Rules

Rule	Description
"All shapes have at least one line of symmetry"	Every shape in the box has reflective symmetry (e.g., circles, squares, equilateral triangles — but not scalene triangles or parallelograms)
"At least one shape has no line of symmetry"	Every box contains at least one asymmetric shape
"All shapes have rotational symmetry of order 2 or more"	Every shape looks the same after a 180° rotation

Worked Example: All Shapes Have Line Symmetry

Set A:

• Box 1: Circle, square, equilateral triangle (all have line symmetry ✓)
• Box 2: Regular pentagon, isosceles triangle, rectangle (all have line symmetry ✓)
• Box 3: Regular hexagon, circle, square (all have ✓)
• Box 4: Equilateral triangle, circle, regular pentagon (all have ✓)
• Box 5: Square, rectangle, isosceles triangle (all have ✓)
• Box 6: Circle, circle, regular hexagon (all have ✓)

Set B:

• Box 1: Parallelogram, circle, square (parallelogram has 0 lines of symmetry ✗)
• Box 2: Scalene triangle, hexagon, rectangle (scalene triangle has 0 ✗)
• Box 3: Irregular quadrilateral, pentagon, circle (irregular quad has 0 ✗)
• Box 4: Parallelogram, parallelogram, triangle (parallelograms have 0 ✗)
• Box 5: Scalene triangle, circle (scalene has 0 ✗)
• Box 6: Irregular pentagon, square, hexagon (irregular pentagon has 0 ✗)

Rule A: Every shape has at least one line of symmetry. Rule B: At least one shape has no line of symmetry.

Test shape: A circle, a rectangle, and a parallelogram.

• Circle: has line symmetry ✓
• Rectangle: has line symmetry ✓
• Parallelogram: has no line of symmetry ✗
• Not all shapes have line symmetry → Does not fit Set A
• Contains a shape with no line of symmetry → Fits Set B
• Answer: Set B

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Category 2: Arrangement Symmetry Rules

The rule is not about individual shapes but about whether the overall arrangement of shapes in the box is symmetrical.

Worked Example: The Box Has Vertical Line Symmetry

Set A:

• Box 1: A triangle at the top-centre; a circle bottom-left and a circle bottom-right (mirror image arrangement about the vertical centre line)
• Box 2: A square in the centre; a pentagon top-left and a pentagon top-right
• Box 3: Three shapes in a row across the centre (left shape mirrors right shape)
• Box 4: A diamond at the top; two identical shapes at the bottom-left and bottom-right

Rule: The arrangement of shapes in each box is symmetrical about the vertical centre line. Shapes on the left side of the box have corresponding shapes on the right side.