4.2 Polygons

Learning objectives

4.2A Names of polygonsRecognise and name parallelograms, rectangles, squares, rhombuses, trapeziums, kites, pentagons, hexagons and octagons.4.2B Quadrilaterals and their angle sumunderstand and use the term ‘quadrilateral’ and the angle sum property of quadrilaterals The four angles of a quadrilateral are 90°, (x + 15)°, (x +25)° and (x + 35)° Find the value of x 4.2C Properties of quadrilateralsunderstand and use the properties of the parallelogram, rectangle, square, rhombus, trapezium and kite 4.2D Regular polygons and their anglesunderstand the term ‘regular polygon’ and calculate interior and exterior angles of regular polygons 4.2E Angle sum of polygonsunderstand and use the angle sum of polygons For a polygon with n sides, the sum of the interior angles is (2n – 4) right angles 4.2F Congruence as meaning the same shape and sizeunderstand congruence as meaning the same shape and size 4.2G Congruent polygonsUnderstand that polygons with the same shape and size are congruent.

A polygon is a closed 2D shape made only from straight line segments. Name it first by its number of sides, then use special properties when a more specific quadrilateral name applies.

Sides	General name
4	quadrilateral
5	pentagon
6	hexagon
8	octagon

Parallelogram, rectangle, square, rhombus, trapezium and kite are all quadrilaterals distinguished by side, angle and parallel-line properties.

A circle is not a polygon because its boundary is curved. A shape must be closed; disconnected or open line segments do not form a polygon.

A quadrilateral has four sides and four interior angles. Its interior angles always sum to $360^\circ$ .

Step	Action
1	identify the four interior angles
2	add their values or algebraic expressions
3	set the total equal to $360^\circ$
4	solve and substitute back to check

If the angles are $90^\circ$ , $(x+15)^\circ$ , $(x+25)^\circ$ and $(x+35)^\circ$ , then $90+x+15+x+25+x+35=360$ , giving $x=65$ .

Use interior angles only. An exterior angle shown beside a vertex must first be converted using the straight-line sum if appropriate.

Classify a quadrilateral from guaranteed properties, not visual appearance. Parallel arrows, equal-side ticks and right-angle squares carry exact information.

Shape	Key properties
parallelogram	opposite sides parallel and equal; opposite angles equal
rectangle	four right angles; opposite sides equal and parallel
square	four equal sides and four right angles
rhombus	four equal sides; opposite sides parallel
trapezium	one pair of parallel sides
kite	two pairs of adjacent equal sides

A rectangle has equal diagonals that bisect each other; a rhombus has perpendicular diagonals that bisect each other; a square has both sets of properties.

A square is also a rectangle, rhombus and parallelogram. Classification categories can overlap when one shape satisfies another's definition.

A regular polygon has all sides equal and all interior angles equal. Its exterior angles are also equal.

Quantity for a regular $n$ -gon	Formula
each exterior angle	$360^\circ/n$
each interior angle	$180^\circ-360^\circ/n$
number of sides from exterior angle $e$	$n=360^\circ/e$

If each exterior angle is $24^\circ$ , then $n=360/24=15$ . If each interior angle is $162^\circ$ , the exterior angle is $18^\circ$ , so $n=20$ .

The $360^\circ$ division applies to one exterior angle of a regular polygon. Interior angles do not generally sum to $360^\circ$ .

For any $n$ -sided polygon, the sum of interior angles is $(n-2)\times180^\circ$ , equivalent to $(2n-4)$ right angles.

Polygon	$n$	Interior-angle sum
triangle	3	$180^\circ$
quadrilateral	4	$360^\circ$
pentagon	5	$540^\circ$
decagon	10	$1440^\circ$

Subtract all known interior angles from the total to find a missing angle. For algebraic angles, form one equation equal to the total.

Drawing diagonals from one vertex divides an $n$ -gon into $n-2$ triangles, which explains the formula.

This formula gives the sum for both regular and irregular polygons. Divide by $n$ only when the polygon is regular and each angle is equal.

Two figures are congruent when one can be placed exactly on the other using translations, rotations or reflections. Corresponding lengths and angles are equal.

Relationship	Same shape?	Same size?
congruent	yes	yes
similar but not congruent	yes	not necessarily
equal area only	not necessarily	not enough information

A congruent copy may face a different direction or be reflected. Orientation and position do not change length or angle measurements.

Match vertices in order and compare every corresponding side and angle. A single mismatch proves the figures are not congruent.

Same area or same perimeter alone does not prove congruence; different shapes can share either measurement.

Trace the vertices of one polygon in order, then find an ordering of the other polygon with the same sequence of side lengths and included angles.

Step	Check
1	same number of sides
2	matching side-length pattern in order
3	matching angle pattern in order
4	allow rotation, translation or reflection

A congruence statement must list corresponding vertices in matching order. If $ABCD$ matches $PQRS$ , then $AB$ corresponds to $PQ$ and angle $B$ to angle $Q$ .

Looking similar is insufficient. A scaled copy has the same angle pattern but different side lengths, so it is similar rather than congruent.

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—4.3 SymmetryRead

4.2 Polygons

Learning objectives

Recognise and name common polygons

Use the quadrilateral angle sum

Use properties of special quadrilaterals

Calculate angles in regular polygons

Use the interior-angle sum of any polygon

Understand congruence as same shape and size

Identify congruent polygons by correspondence

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