Types of Quadrilaterals

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In this lesson, you will learn

  • The six common types of quadrilaterals and how to tell them apart
  • Which sides are parallel or equal, and which angles are equal, in each type
  • The three types of trapezoids
  • How to name any quadrilateral by asking a few questions about its sides and angles
Definition

A quadrilateral is a polygon with four sides, four angles and four vertices.

There are six common types of quadrilaterals, and these quadrilaterals can be classified by the relationships of their sides and angles.

How to Identify a Parallelogram

Definition

A parallelogram is a four-sided polygon with two pairs of parallel and equal sides.

The following figure is a parallelogram.

Parallelogram ABCD: side AB is parallel to side DC, and side AD is parallel to side BC

In a parallelogram, all the following are true:

  • Opposite sides are parallel (AB || DC and AD || BC)
  • Opposite sides are equal (AB = DC and AD = BC)
  • Opposite angles are equal (angle A = angle C and angle B = angle D)

In parallelogram ABCD, angle B = 115°. What is angle D?

Show answer

Angle D = 115°. Angles B and D are opposite angles, and opposite angles in a parallelogram are equal.

How to Identify a Rectangle

Definition

A rectangle is a parallelogram with 4 right angles.

The following figure is a rectangle.

Rectangle EFGH: HG is parallel to EF, HE is parallel to GF, and all four angles are 90 degrees

In a rectangle, all the following are true:

  • Opposite sides are parallel (HG || EF and HE || GF)
  • Opposite sides are equal (HG = EF and HE = GF)
  • All four angles are equal to 90 degrees (angle E = angle F = angle G = angle H = 90 degrees)

Is every rectangle also a parallelogram?

Show answer

Yes. A rectangle has two pairs of parallel and equal sides, so it is a parallelogram. It is a special one, because all four of its angles are right angles.

How to Identify a Square

Definition

A square is a rectangle with 4 equal sides.

The following figure is a square.

Square IJKL: JI is parallel to KL, JK is parallel to IL, all four sides are equal and all four angles are 90 degrees

In a square, all the following are true:

  • Opposite sides are parallel (JK || IL and JI || KL)
  • All four sides are equal (JI = KL = JK = IL)
  • All four angles are equal to 90 degrees (angle I = angle J = angle K = angle L = 90 degrees)
Common mistake

Thinking that a square is not a rectangle. Every square is a rectangle, because it is a parallelogram with 4 right angles. But not every rectangle is a square: a rectangle only needs its opposite sides to be equal.

A rectangle has sides of 6 cm, 4 cm, 6 cm and 4 cm. Is it a square?

Show answer

No. A square needs all four sides to be equal. Here two sides are 6 cm and two sides are 4 cm.

How to Identify a Rhombus

Definition

A rhombus is a parallelogram with 4 equal sides.

The following figure is a rhombus.

Rhombus MNOP: PO is parallel to MN, MP is parallel to NO, and all four sides are equal

In a rhombus, all the following are true:

  • Opposite sides are parallel (PO || MN and PM || ON)
  • All four sides are equal (PO = ON = NM = MP)
  • Opposite angles are equal (angle M = angle O and angle N = angle P)
Remember

A square and a rhombus both have 4 equal sides. The difference is the angles: a square has 4 right angles, while a rhombus does not need to have any.

In rhombus MNOP, side PO = 9 cm. How long is side NM?

Show answer

NM = 9 cm. All four sides of a rhombus are equal.

How to Identify a Trapezoid

Definition

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

There are three common types of trapezoids.

Scalene trapezoid

A scalene trapezoid is a trapezoid with no equal sides. The following is a scalene trapezoid.

Scalene trapezoid QRST: RS is parallel to QT and no two sides are equal

Right-angled trapezoid

A right-angled trapezoid is a trapezoid with two right angles. The following is a right-angled trapezoid.

Right-angled trapezoid QRST: RS is parallel to QT, and angle R and angle Q are both 90 degrees

Isosceles trapezoid

In an isosceles trapezoid, the non-parallel sides are equal. The following is an isosceles trapezoid.

Isosceles trapezoid QRST: RS is parallel to QT, and the non-parallel sides RQ and ST are equal

A quadrilateral has exactly one pair of parallel sides, and its two non-parallel sides are both 5 cm long. What type of quadrilateral is it?

Show answer

An isosceles trapezoid. One pair of parallel sides makes it a trapezoid, and equal non-parallel sides make it isosceles.

How to Identify a Kite

Definition

A kite is a quadrilateral with two pairs of adjacent sides that are equal. Furthermore, no opposite sides are equal.

Kite UVWX: the two top sides UX and UV are equal, and the two bottom sides WX and WV are equal
Common mistake

Looking for equal opposite sides in a kite. In a kite, the equal sides are next to each other (adjacent): UX = UV and WX = WV. If the opposite sides are equal, the shape is a parallelogram, not a kite.

In kite UVWX, UX = 4 cm and WV = 9 cm. How long are UV and WX?

Show answer

UV = 4 cm and WX = 9 cm. UV is next to UX and equal to it, and WX is next to WV and equal to it.

How to Identify Any Quadrilateral

To name a quadrilateral, ask these questions in order.

  1. How many pairs of parallel sides does it have? Two pairs: it is a parallelogram (go to question 2). One pair: it is a trapezoid (go to question 3). None: check whether it is a kite (two pairs of equal adjacent sides).
  2. For a parallelogram, look at the angles and sides. 4 right angles and 4 equal sides: square. 4 right angles only: rectangle. 4 equal sides only: rhombus. Neither: parallelogram.
  3. For a trapezoid, look at the non-parallel sides. Equal: isosceles trapezoid. Two right angles: right-angled trapezoid. No equal sides: scalene trapezoid.
Worked example: A quadrilateral has two pairs of parallel sides. All four sides are 5 cm long, and two of its angles are 60° and 120°. What type of quadrilateral is it?
  1. Count the pairs of parallel sides: there are two pairs, so it is a parallelogram.
  2. Check the angles: 60° and 120° are not right angles, so it is not a rectangle or a square.
  3. Check the sides: all four sides are 5 cm, so the sides are equal.
  4. A parallelogram with 4 equal sides and no right angles is a rhombus.

Answer: it is a rhombus.

A quadrilateral has no parallel sides. Its two top sides are 3 cm each, and its two bottom sides are 8 cm each. What type of quadrilateral is it?

Show answer

A kite. It has two pairs of equal adjacent sides, and no opposite sides are equal.

Summary

  • A parallelogram has two pairs of parallel and equal sides, and its opposite angles are equal.
  • A rectangle is a parallelogram with 4 right angles. A rhombus is a parallelogram with 4 equal sides.
  • A square is a rectangle with 4 equal sides, so it is also a rhombus and a parallelogram.
  • A trapezoid has only one pair of parallel sides. It can be scalene, right-angled or isosceles.
  • A kite has two pairs of equal adjacent sides, and no opposite sides are equal.

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