Quadrilaterals

A has four sides and four angles. The five P5 shapes in this lesson are related: some are special cases of others. Name the properties first, then use the angle rule that fits the figure.

Parents: ask your child to say the shape and the property before revealing an answer. The lab lets them turn each visible property mark on and off.

By the end you will be able to recognise the five quadrilaterals, describe their sides and angles, and find unknown angles without adding an extra line to the figure.

Keep the shape relationships inclusive

Use the full property list when classifying a shape. A shape can belong to more than one family:

Shape	Properties to check	Relationship
Square	Four equal sides and four right angles	A square is both a rectangle and a rhombus, so it is also a parallelogram.
Rectangle	Opposite sides equal and parallel; four right angles	Every rectangle is a parallelogram. A rectangle need not have four equal sides.
Rhombus	Four equal sides; opposite sides parallel; opposite angles equal	Every rhombus is a parallelogram. A rhombus need not have right angles.
Parallelogram	Two pairs of opposite parallel sides; opposite sides and angles equal	It has two pairs of parallel sides.
Trapezium	Exactly one pair of parallel sides	This lesson uses the “exactly one pair” convention, so a parallelogram is not called a trapezium here.

The word opposite means across the shape. The word neighbouring means next to each other along one side. These words matter when you describe a parallelogram.

🤔 Predict first: Which shape has four equal sides and four right angles?

See the properties change the drawing

Quadrilateral property lab

Predict first: Which shape has four equal sides and four right angles?

The shape buttons use verified coordinate fixtures. The marks are drawn from the actual sides and vertices: equal-side ticks sit on their side, parallel arrows sit on the parallel sides, right-angle corners use perpendicular vectors, and angle labels are calculated from the two sides meeting at each vertex. Turn on one property at a time and read the consequence below the figure.

For example, the square in the lab has four sides of the same length and four angles of $90^\circ$ . It therefore satisfies both the rectangle and rhombus definitions. The rhombus preset has equal sides but angles of $60^\circ$ and $120^\circ$ , so it shows why “four equal sides” does not mean “four right angles”.

Given and unknown angles

The lower lab panel gives one angle and marks the unknown as $x$ . It uses the shown sides and vertices only: no diagonal or extension is added.

In the figure, angle A is 45° and the opposite angle C is x. Find x before opening the check.

Reveal the parallelogram angle check

The fixture gives $\angle A=45^\circ$ and asks for the opposite $\angle C=x$ . Opposite angles in a parallelogram are equal.

Opposite angles

\angle C=\angle A=45^\circ

x=45^\circ

In the figure, rhombus ABCD has four equal sides, angle A is 45°, and the opposite angle C is x. Find x before opening the check.

Reveal the rhombus angle check

The fixture is a rhombus with four equal sides but no right-angle claim. It gives $\angle A=45^\circ$ and asks for the opposite $\angle C=x$ . A rhombus is a parallelogram, so its opposite angles are equal.

Rhombus opposite angles

\angle C=\angle A=45^\circ

x=45^\circ

In the figure, trapezium ABCD has exactly one parallel pair, angle B is 45°, and the same-leg angle C is x. Find x before opening the check.

Reveal the trapezium angle check

Using this lesson's exactly-one-pair convention, the fixture gives $\angle B=45^\circ$ and asks for the same-leg angle $\angle C=x$ . The marked parallel sides make these same-side interior angles supplementary.

Same-side interior angles

\angle B+x=180^\circ

45^\circ+x=180^\circ

x=135^\circ

The angle sum of a quadrilateral

The four interior angles of every quadrilateral add to $360^\circ$ . This includes ordinary quadrilaterals and all five special shapes.

If three angles are known, subtract their total from $360^\circ$ . For angles $82^\circ$ , $96^\circ$ and $104^\circ$ , the unknown angle is $78^\circ$ .

Use the 360-degree sum

82^\circ+96^\circ+104^\circ=282^\circ

360^\circ-282^\circ=78^\circ

\text{unknown angle}=78^\circ

Use parallel sides in a parallelogram

In a parallelogram, opposite angles are equal. Neighbouring angles along one side add to $180^\circ$ . A rhombus has the same angle relationships because it is a parallelogram.

🤔 Predict first: A parallelogram has one angle of 68°. What is the angle next to it along a side?

Show the worked solution

If one angle is $68^\circ$ , the opposite angle is $68^\circ$ . Each neighbouring angle is $112^\circ$ because $180^\circ-68^\circ=112^\circ$ . Use the side and angle labels already given; do not draw an extra line.

Use the one parallel pair in a trapezium

In the trapezium convention used here, the top and bottom sides are the one parallel pair. The two angles between those parallel sides on the same slanted leg are supplementary, so they add to $180^\circ$ .

If a given angle on a slanted leg is $65^\circ$ , its partner is $115^\circ$ . This uses the parallel-side relationship in the figure. It does not assume that the two non-parallel sides are equal.

Same slanted leg

65^\circ+x=180^\circ

x=180^\circ-65^\circ=115^\circ

Graduated practice

Try each complete question before opening its collapsed solution. Keep the given properties in view while you decide which rule applies.

Practice 1 — classify from properties. A quadrilateral has four equal sides and four right angles. Name the most specific shape, then name the two special shape families it also belongs to.

Reveal solution: Practice 1

It is a square. It is also a rectangle because it has four right angles, and a rhombus because it has four equal sides. Both are parallelograms.

Classify the shape

\text{four equal sides}+\text{four right angles}\Rightarrow\text{square}

\text{square}\subset\text{rectangle and rhombus}

Practice 2 — use the 360° sum. A quadrilateral has angles $74^\circ$ , $88^\circ$ , $103^\circ$ and $x$ . Find $x$ .

Reveal solution: Practice 2

Find the fourth angle

74^\circ+88^\circ+103^\circ=265^\circ

x=360^\circ-265^\circ=95^\circ

Practice 3 — use the parallel pair. A trapezium has exactly one pair of parallel sides. One angle at the left slanted side is $71^\circ$ . The angle on the same slanted side between the parallel sides is $x$ . Find $x$ .

Reveal solution: Practice 3

The two angles between the parallel sides on the same slanted side add to $180^\circ$ .

Use the one parallel pair

71^\circ+x=180^\circ

x=180^\circ-71^\circ=109^\circ

Watch out for these mix-ups

Quick recap

🎯 Mastery check

Answer all 8 — your progress is saved on this device.

Which shape has four equal sides and four right angles?
A rectangle has opposite sides equal and how many right angles?
A rhombus has an angle of 45°. What is the opposite angle?
How many pairs of parallel sides does a parallelogram have?
Using this lesson’s convention, how many pairs of parallel sides does a trapezium have?
A quadrilateral has angles 74°, 88°, 103° and x. What is x?
A parallelogram has one angle of 68°. What is the angle next to it along a side?
A trapezium has one angle of 71° on a slanted side. What is the angle between the same parallel sides on that side?