Year 7 Maths Geometry

Classifying Quadrilaterals

Q: What is a non-convex quadrilateral?

A quadrilateral with one reflex angle (more than 180°), so it has a dent.

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Theory

A quadrilateral is a closed shape with four straight sides and four vertices. The special quadrilaterals (trapezium, parallelogram, rhombus, rectangle, square and kite) are sorted by their sides and angles. This Year 7 maths page shows how to name a quadrilateral, read the marks on a diagram and give the most exact name.

Naming. Write the vertices in order around the shape, starting anywhere and going either way: \(ABCD\), \(BCDA\) and \(ADCB\) name the same shape. \(ACBD\) does not, because \(A\) to \(C\) jumps across a diagonal (a line joining two vertices that are not next to each other).

Marks. Tick marks show equal sides. Arrow marks show parallel sides. A small square shows a right angle.

The special quadrilaterals.
Trapezium: at least one pair of parallel sides.
Parallelogram: both pairs of opposite sides parallel (and equal).
Rhombus: a parallelogram with all four sides equal.
Rectangle: a parallelogram with four right angles.
Square: a rectangle with all four sides equal.
Kite: two pairs of adjacent (next-door) sides equal.

More than one name. A square is also a rectangle, a rhombus, a parallelogram, a kite and a trapezium. Give the most exact name: the one that tells you the most.

Convex or non-convex. In a convex quadrilateral every angle is less than \(180^\circ\). A non-convex (concave) quadrilateral has one reflex angle (more than \(180^\circ\)), so it looks pushed in, like a dent.

The six special quadrilaterals with their marks: arrows for parallel sides, ticks for equal sides and small squares for right angles.

Each arrow points to a special kind of the shape above it. Every square is a rectangle and a rhombus; every rhombus is a parallelogram and a kite.

A non-convex quadrilateral has a reflex angle at its dent (here \(230^\circ\) at \(C\)). In a convex quadrilateral every angle is less than \(180^\circ\).

What each special quadrilateral must have:

Shape	Sides	Angles
trapezium	at least one pair parallel	—
parallelogram	opposite sides parallel and equal	opposite angles equal
rhombus	all four equal, opposite sides parallel	opposite angles equal
rectangle	opposite sides parallel and equal	all four \(90^\circ\)
square	all four equal, opposite sides parallel	all four \(90^\circ\)
kite	two pairs of adjacent sides equal	one pair of opposite angles equal

Their diagonals:

Shape	Diagonals
trapezium	—
parallelogram	bisect each other
rhombus	bisect each other at right angles
rectangle	equal, bisect each other
square	equal, bisect each other at right angles
kite	meet at right angles, one is bisected

          Use the marks, not the look. A shape that looks like a square is only a square if the marks show four equal sides and four right angles.
        

          The family only works one way. Every square is a rectangle, but not every rectangle is a square. Every rhombus is a parallelogram, but not every parallelogram is a rhombus.
        

How to classify a quadrilateral

Count the pairs of parallel sides (arrow marks).
Check the sides: equal opposite sides, all four equal, or two pairs of adjacent sides equal (tick marks).
Check the angles: right-angle marks, or an angle more than \(180^\circ\).
Choose the most exact name that fits every mark.

How to name a quadrilateral

Start at any vertex.
Go around the shape to the next vertex, either way, until all four are used.
Never jump across a diagonal.

Example 1 — Naming a quadrilateral

Which of these are names for quadrilateral \(KLMN\)?
(i) \(LMNK\)
(ii) \(KNML\)
(iii) \(KMLN\)

Solution

(i) Yes: start at \(L\) and go around.
(ii) Yes: go around the other way.
(iii) No: \(K\) to \(M\) is the diagonal (dashed), not a side.

Example 2 — Read the marks

Give the most exact name for each shape.

Solution

(i) Four right angles, but the one-tick and two-tick sides are different lengths: a rectangle, not a square.
(ii) Four equal sides and no right angles: a rhombus.

Example 3 — Convex or non-convex

The angles of quadrilateral \(ABCD\) are \(70^\circ\), \(30^\circ\), \(230^\circ\) and \(30^\circ\). Is it convex or non-convex?

Solution

\(70 + 30 + 230 + 30\)

\(=\)

\(360\)

230 > 180, so one angle is reflex non-convex

\(230^\circ > 180^\circ\), so one angle is reflex. \(ABCD\) is non-convex (see the shape in Diagrams).

Example 4 — On the number plane

\(A(1, 3)\), \(B(4, 5)\), \(C(7, 3)\) and \(D(4, 1)\). Find the most exact name for \(ABCD\), and two other names.

Solution

Count the steps along each side. \(AB\) and \(DC\): \(3\) right, \(2\) up. \(AD\) and \(BC\): \(3\) right, \(2\) down. The same steps mean \(AB \parallel DC\) and \(AD \parallel BC\), and every side is \(3\) across and \(2\) up or down, so all four sides are equal.

It is not a square: \(ABCD\) is \(6\) units wide (\(A\) to \(C\)) but only \(4\) units tall (\(D\) to \(B\)). It is a rhombus (also a parallelogram and a kite).

Example 5 — Clues about the diagonals and angles

Which quadrilateral must it be?
(i) Its diagonals bisect each other at right angles, but its angles are not all equal.
(ii) Both pairs of opposite angles are equal, but its sides are not all equal and its angles are not all equal.

Solution

(i) Only a rhombus and a square have diagonals that bisect each other at right angles. The angles are not all equal, so it is not a square: a rhombus.
(ii) A parallelogram, rhombus, rectangle and square all have equal opposite angles. Not all angles equal rules out the rectangle and square; not all sides equal rules out the rhombus: a parallelogram.

Example 6 — One clue at a time

Clues about quadrilateral \(WXYZ\) are given one at a time. After each clue, give the most exact name.
(i) All four angles are \(90^\circ\).
(ii) As well: \(WX = XY\).

Solution

(i) Four right angles: a rectangle.
(ii) A rectangle with two adjacent sides equal has all four sides equal: a square.

Common pitfalls

Naming across a diagonal. \(WYXZ\) is not a name for \(WXYZ\): \(W\) to \(Y\) is a diagonal.

Not the most exact name. A square is a parallelogram, but "square" tells you more, so it is the best answer.

Kites and parallelograms. A kite has equal sides next to each other; a parallelogram has equal sides opposite each other.

Trusting the picture. Only the marks (or measurements) prove that sides are equal or parallel.

Frequently asked questions

What is a quadrilateral?

A closed shape with four straight sides and four vertices.

What is the difference between a rhombus and a square?

Both have four equal sides. A square also has four right angles.

Is a square a rectangle?

Yes. A square has four right angles and opposite sides parallel, so it is a special rectangle.

What is a kite?

A quadrilateral with two pairs of equal sides that are next to each other (adjacent).

What is a non-convex quadrilateral?

A quadrilateral with one reflex angle (more than \(180^\circ\)), so it has a dent.

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Exterior Angle of a Triangle

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Angle Sum of a Quadrilateral

Classifying Quadrilaterals

📖 Theory

How to classify a quadrilateral

How to name a quadrilateral

Common pitfalls

Frequently asked questions

Theory