4. Quadrilaterals

What Is a Quadrilateral?

A quadrilateral is a polygon with:

• 4 straight sides
• 4 vertices
• 4 interior angles

The word comes from terms meaning four and side.

There are many types of quadrilaterals. They are classified according to properties such as:

• Equal side lengths.
• Parallel sides.
• Perpendicular sides.
• Equal angles.
• Right angles.
• Diagonals.

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Interior Angles of a Quadrilateral

The interior angles of every quadrilateral add to:

360°

This can be understood by dividing a quadrilateral into two triangles.

Each triangle has:

180°

Therefore:

180° + 180° = 360°

This rule applies to every quadrilateral, regardless of its shape.

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Finding a Missing Angle

Suppose a quadrilateral has angles:

90°, 80°, 110°, and x

Since the total is 360°:

90° + 80° + 110° + x = 360°

Add the known angles:

280° + x = 360°

Therefore:

x = 80°

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Parallel Sides

Parallel sides remain the same distance apart and never meet if extended.

The symbol for parallel is:

∥

For example:

AB ∥ CD

means that AB is parallel to CD.

Parallel sides are an important feature used to classify quadrilaterals.

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Perpendicular Sides

Perpendicular sides meet at:

90°

The symbol for perpendicular is:

⊥

For example:

AB ⊥ BC

means that AB and BC meet at a right angle.

Squares and rectangles have perpendicular adjacent sides.

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Squares

A square is a quadrilateral with:

• 4 equal sides.
• 4 right angles.
• 2 pairs of parallel sides.

A square has:

4 equal sides

and:

4 angles of 90°

Since:

4 × 90° = 360°

its interior angles satisfy the quadrilateral angle rule.

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Properties of a Square

A square has:

• Four equal sides.
• Four equal angles.
• Four right angles.
• Two pairs of parallel sides.
• Adjacent sides that are perpendicular.
• Opposite sides that are equal.
• Two diagonals of equal length.
• Diagonals that bisect each other.
• Diagonals that meet at 90°.
• Four lines of symmetry.

A square has many properties because it belongs to several different quadrilateral categories.

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Rectangles

A rectangle is a quadrilateral with:

• 4 right angles.
• Opposite sides equal.
• 2 pairs of parallel sides.

A rectangle does not need to have all four sides equal.

For example, a rectangle might measure:

8 cm × 5 cm

Opposite sides are equal:

8 cm and 8 cm

5 cm and 5 cm

and every angle is:

90°

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Properties of a Rectangle

A rectangle has:

• Four right angles.
• Opposite sides equal.
• Opposite sides parallel.
• Adjacent sides perpendicular.
• Two diagonals of equal length.
• Diagonals that bisect each other.

A rectangle usually has:

2 lines of symmetry

A square is a special rectangle with four equal sides.

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Parallelograms

A parallelogram is a quadrilateral with:

2 pairs of parallel opposite sides

Its opposite sides are also equal in length.

A parallelogram does not need to have right angles.

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Properties of a Parallelogram

A parallelogram has:

• Two pairs of parallel sides.
• Opposite sides equal.
• Opposite angles equal.
• Adjacent angles that add to 180°.
• Diagonals that bisect each other.

For example, if one angle is:

65°

the opposite angle is:

65°

Each adjacent angle is:

180° − 65° = 115°

Therefore, its angles are:

65°, 115°, 65°, 115°

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Rhombuses

A rhombus is a quadrilateral with:

4 equal sides

Opposite sides are parallel.

Unlike a square, a rhombus does not need to have four right angles.

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Properties of a Rhombus

A rhombus has:

• Four equal sides.
• Two pairs of parallel sides.
• Opposite angles equal.
• Adjacent angles that add to 180°.
• Diagonals that bisect each other.
• Diagonals that meet at 90°.

A square is therefore a special type of rhombus.

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Trapezoids

A trapezoid is a quadrilateral with at least one pair of parallel sides.

In some curricula, the term is defined as having exactly one pair of parallel sides. It is useful to check which convention your course uses.

The parallel sides of a trapezoid are often called its:

bases

The non-parallel sides are often called its:

legs

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Isosceles Trapezoids

An isosceles trapezoid has equal non-parallel sides.

It also has pairs of equal angles.

For example, its angles might be:

70°, 70°, 110°, 110°

These add to:

360°

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Comparing the Main Quadrilaterals

*This table uses the inclusive definition of trapezoid: at least one pair of parallel sides.

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A Family of Quadrilaterals

An important idea in geometry is that quadrilateral categories can overlap.

A shape can belong to more than one category.

For example, a square satisfies the definition of:

• A quadrilateral.
• A parallelogram.
• A rectangle.
• A rhombus.

This creates a hierarchy of shapes.

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Every Square Is a Rectangle

A rectangle must have:

• Four right angles.
• Two pairs of parallel sides.
• Opposite sides equal.

A square has all of these properties.

Therefore:

Every square is a rectangle.

However, a rectangle does not need four equal sides.

Therefore:

Not every rectangle is a square.

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Every Square Is a Rhombus

A rhombus must have:

four equal sides

A square has four equal sides.

Therefore:

Every square is a rhombus.

However, a rhombus does not need four right angles.

Therefore:

Not every rhombus is a square.

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Rectangles Are Parallelograms

A parallelogram must have:

two pairs of parallel opposite sides

Every rectangle has two pairs of parallel opposite sides.

Therefore:

Every rectangle is a parallelogram.

But not every parallelogram has right angles.

Therefore:

Not every parallelogram is a rectangle.

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Rhombuses Are Parallelograms

Every rhombus has:

two pairs of parallel opposite sides

Therefore:

Every rhombus is a parallelogram.

But parallelograms do not need four equal sides.

Therefore:

Not every parallelogram is a rhombus.

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The Special Position of the Square

A square combines the defining properties of both:

a rectangle

and:

a rhombus

It has:

four equal sides

and:

four right angles

This makes the square a particularly important quadrilateral.

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Classification by Parallel Sides

Parallel sides can help identify a quadrilateral.

Two pairs of parallel sides

The shape could be:

• Parallelogram.
• Rectangle.
• Rhombus.
• Square.

More information is needed to classify it further.

One pair of parallel sides

Using the exclusive definition, the shape is a:

trapezoid

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Classification by Equal Sides

Suppose a quadrilateral has:

four equal sides

It could be:

• A rhombus.
• A square.

Now examine its angles.

If all angles are:

90°

it is a square.

If the angles are not all 90°, it is a rhombus but not a square.

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Classification by Right Angles

Suppose a quadrilateral has:

four right angles

It must be a:

rectangle

If it also has:

four equal sides

then it is more specifically a:

square

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Classification by Parallel Sides and Angles

Suppose a quadrilateral has:

• Two pairs of parallel sides.
• Opposite sides equal.
• No right angles.
• Not all four sides equal.

It is a:

parallelogram

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Using Side Markings in Diagrams

Geometry diagrams often use small marks to show equal lengths.

Sides with the same marking have equal lengths.

For example:

• One tick on two sides means those sides are equal.
• Two ticks on another pair indicate that those two sides are equal to each other.

Different markings do not mean equal lengths.

These symbols allow us to classify shapes even when no measurements are written.

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Using Arrow Markings

Small arrow markings are often used to show parallel sides.

Sides with matching arrow markings are parallel.

For example:

one arrow → one pair of parallel sides

two matching arrow pairs → two pairs of parallel sides

Do not rely only on how parallel the sides appear in a drawing.

Use the markings provided.

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Right-Angle Markings

A small square placed inside a corner means:

90°

If all four corners of a quadrilateral are right angles, the shape is at least a:

rectangle

If all four sides are also equal, it is a:

square

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Diagonals

A diagonal is a line segment connecting two non-adjacent vertices.

Every quadrilateral has:

2 diagonals

Diagonals can provide useful information for classifying quadrilaterals.

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Diagonals of a Square

The diagonals of a square:

• Are equal in length.
• Bisect each other.
• Meet at 90°.
• Bisect the vertex angles.

This combination of properties is especially useful for identifying squares.

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Diagonals of a Rectangle

The diagonals of a rectangle:

• Are equal in length.
• Bisect each other.

They do not generally meet at 90°.

A square is the special case where they do.

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Diagonals of a Rhombus

The diagonals of a rhombus:

• Bisect each other.
• Meet at 90°.
• Bisect opposite angles.

They are not generally equal in length.

Again, a square is the special case where they are equal.

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Diagonals of a Parallelogram

The diagonals of a parallelogram:

• Bisect each other.

They are not necessarily:

• Equal.
• Perpendicular.

This helps distinguish a general parallelogram from rectangles and rhombuses.

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Solving Angle Problems in Parallelograms

Suppose one angle of a parallelogram is:

72°

The opposite angle is also:

72°

An adjacent angle is:

180° − 72°

= 108°

Therefore, the four angles are:

72°, 108°, 72°, 108°

Check:

72 + 108 + 72 + 108 = 360°

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Solving an Algebraic Angle Problem

Suppose adjacent angles of a parallelogram are:

x

and:

2x

Adjacent angles add to:

180°

Therefore:

x + 2x = 180°

3x = 180°

x = 60°

The other angle is:

2x = 120°

Therefore, the parallelogram has angles:

60°, 120°, 60°, 120°

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Finding Missing Sides

Opposite sides of a parallelogram are equal.

Suppose a parallelogram has:

top side = 12 cm

left side = 7 cm

Then:

bottom side = 12 cm

and:

right side = 7 cm

Its perimeter is:

12 + 7 + 12 + 7

= 38 cm

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Perimeter of a Square

A square has four equal sides.

Therefore:

P = 4s

Suppose:

s = 8 cm

Then:

P = 4 × 8

P = 32 cm

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Perimeter of a Rectangle

For a rectangle:

P = 2l + 2w

Suppose:

l = 10 cm

w = 6 cm

Then:

P = 2(10) + 2(6)

P = 32 cm

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Area of Squares and Rectangles

For a rectangle:

A = l × w

For a square:

A = s²

Example

A rectangle measures:

12 cm × 5 cm

Therefore:

A = 12 × 5

A = 60 cm²

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Area of a Parallelogram

The area of a parallelogram is:

A = b × h

where:

b = base

and:

h = perpendicular height

The perpendicular height is not necessarily the length of the sloping side.

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Area of a Trapezoid

The area of a trapezoid can be calculated using the lengths of its parallel sides and its perpendicular height.

If the parallel sides are:

a and b

and the perpendicular height is:

h

then:

A = ½(a + b)h

Example

Suppose:

a = 8 cm

b = 12 cm

h = 5 cm

Then:

A = ½(8 + 12)(5)

A = ½(20)(5)

A = 50 cm²

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Quadrilaterals in the Real World

Quadrilaterals are extremely common in everyday objects and structures.

Examples include:

• Doors.
• Windows.
• Screens.
• Books.
• Tables.
• Floor tiles.
• Road signs.
• Building designs.
• Picture frames.
• Roof structures.

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Quadrilaterals in Engineering and Design

Different quadrilateral properties can be useful for different purposes.

Rectangles are common because:

• They fit together without gaps.
• They provide right-angle corners.
• Their dimensions are easy to measure.

Parallelograms can appear in:

• Mechanical linkages.
• Folding mechanisms.
• Scissor mechanisms.

Trapezoidal shapes may be used in:

• Bridges.
• Roof structures.
• Containers.
• Structural designs.

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Comparing Quadrilaterals

When comparing quadrilaterals, examine:

Sides

• Are any equal?
• Are all equal?

Parallel relationships

• Is there one pair?
• Are there two pairs?

Angles

• Are any 90°?
• Are opposite angles equal?

Diagonals

• Are they equal?
• Are they perpendicular?
• Do they bisect each other?

Using several properties together allows a shape to be classified accurately.

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A Classification Strategy

Suppose you are given an unknown quadrilateral.

Ask: Are there two pairs of parallel sides?

If yes, it belongs to the parallelogram family.

Then ask:

Are all four angles 90°?

If yes, it is a rectangle.

Then ask:

Are all four sides equal?

If yes, it is a square.

Alternatively, if all four sides are equal but the angles are not all 90°, it is a rhombus.

If neither special condition applies, it is a general parallelogram.

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Most Specific Classification

Sometimes a shape fits several categories.

Suppose a quadrilateral has:

• Four equal sides.
• Four right angles.
• Two pairs of parallel sides.

It is technically:

• A quadrilateral.
• A parallelogram.
• A rectangle.
• A rhombus.
• A square.

When asked for the most specific classification, the answer is:

square

The broader categories are still mathematically correct.

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Common Mistakes

Thinking a Square Is Not a Rectangle

A rectangle is defined by having four right angles.

A square has four right angles.

Therefore:

a square is a rectangle

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Thinking a Rectangle Must Have Two Long and Two Short Sides

A rectangle requires opposite sides to be equal and all angles to be 90°.

Its adjacent sides can also be equal.

When this happens, the rectangle is a:

square

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Thinking a Rhombus Must Look Like a Diamond

A rhombus is defined by:

four equal sides

Its orientation does not matter.

A square therefore satisfies the definition of a rhombus.

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Classifying Shapes by Appearance

Do not classify a quadrilateral simply because it "looks like" a particular shape.

Use its mathematical properties.

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Assuming Lines Are Parallel

Sides that appear parallel in a sketch may not actually be parallel.

Look for:

• Arrow markings.
• Measurements.
• Stated information.

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Assuming an Angle Is 90°

An angle that looks like a right angle is not necessarily 90°.

Look for a right-angle symbol or other given information.

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Confusing Parallel and Perpendicular

Remember:

Parallel → never meet

Perpendicular → meet at 90°

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Forgetting the Quadrilateral Angle Sum

Every quadrilateral has interior angles adding to:

360°

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Confusing a Rhombus and a Parallelogram

Every rhombus is a parallelogram.

But not every parallelogram has four equal sides.

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Check Your Understanding

1. Define a quadrilateral.

2. How many sides does every quadrilateral have?

3. How many vertices does every quadrilateral have?

4. What is the sum of the interior angles of a quadrilateral?

5. Describe the properties of a square.

6. Describe the properties of a rectangle.

7. Describe the properties of a parallelogram.

8. Describe the properties of a rhombus.

9. Describe a trapezoid.

10. What does it mean for two sides to be parallel?

11. What does it mean for two sides to be perpendicular?

12. Which quadrilaterals have two pairs of parallel sides?

13. Which quadrilateral has four equal sides and four right angles?

14. Which quadrilateral has four equal sides but does not necessarily have right angles?

15. Explain why every square is a rectangle.

16. Explain why not every rectangle is a square.

17. Explain why every square is a rhombus.

18. Explain why every rectangle is a parallelogram.

19. Explain why every rhombus is a parallelogram.

20. A parallelogram has one angle of 70°. Find the other three angles.

21. A quadrilateral has angles 90°, 85°, 105°, and x. Find x.

22. Adjacent angles of a parallelogram are x and 2x. Find both angles.

23. A parallelogram has one side of 12 cm and an adjacent side of 8 cm. Find its perimeter.

24. A square has a perimeter of 36 cm. Find its side length.

25. A rectangle measures 14 cm × 6 cm. Find its perimeter and area.

26. A parallelogram has a base of 10 cm and a perpendicular height of 7 cm. Find its area.

27. A trapezoid has parallel sides of 8 cm and 14 cm and a height of 5 cm. Find its area.

28. A quadrilateral has four equal sides and diagonals that meet at 90°. What classifications could apply?

29. A quadrilateral has four right angles and two pairs of parallel sides. What additional information would you need to determine whether it is a square?

30. Explain how you would classify an unknown quadrilateral using its sides, angles, and parallel relationships.

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Key Terms

• Quadrilateral – polygon with four sides.
• Parallel sides – sides that remain the same distance apart and never meet.
• Perpendicular sides – sides that meet at 90°.
• Square – quadrilateral with four equal sides and four right angles.
• Rectangle – quadrilateral with four right angles.
• Parallelogram – quadrilateral with two pairs of parallel opposite sides.
• Rhombus – quadrilateral with four equal sides.
• Trapezoid – quadrilateral with at least one pair of parallel sides under the inclusive definition.
• Diagonal – line segment joining two non-adjacent vertices.
• Opposite sides – sides that do not share a vertex.
• Adjacent sides – sides sharing a vertex.
• Opposite angles – angles at opposite vertices.
• Adjacent angles – angles sharing a side.
• Bisect – divide into two equal parts.
• Right angle – angle measuring 90°.

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Key Takeaways

• Every quadrilateral has 4 sides, 4 vertices, and 4 interior angles.
• The interior angles of every quadrilateral add to 360°.
• A square has four equal sides and four right angles.
• A rectangle has four right angles and two pairs of parallel sides.
• A parallelogram has two pairs of parallel opposite sides.
• A rhombus has four equal sides and two pairs of parallel sides.
• A trapezoid has at least one pair of parallel sides under the inclusive definition.
• Squares and rectangles have perpendicular adjacent sides.
• Squares, rectangles, rhombuses, and parallelograms all have two pairs of parallel sides.
• Opposite angles of parallelograms are equal.
• Adjacent angles of parallelograms add to 180°.
• Every square is both a rectangle and a rhombus.
• Every rectangle is a parallelogram.
• Every rhombus is a parallelogram.
• Not every rectangle is a square.
• Not every rhombus is a square.
• Quadrilateral categories can overlap because classification is based on properties rather than appearance.

The most important relationship to remember is:

Square → Rectangle and Rhombus → Parallelogram → Quadrilateral

A square does not stop being a rectangle or rhombus simply because square is its more specific name.