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Properties of quadrilaterals

Types of angles Angles on a straight line Angles at a point Vertically opposite angles Angles in parallel lines Collecting like terms Solving equationsThis topic is relevant for:

Here we will learn about **angles in a quadrilateral**, including the sum of angles in a quadrilateral, how to find missing angles, and using these angle facts to generate equations and solve problems.

There are also angles in quadrilaterals worksheets based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if you’re still stuck.

**Angles in a quadrilateral** are the four angles that occur at each vertex within a four-sided shape; these angles are called **interior angles of a quadrilateral**. The sum of the interior angles of any quadrilateral is ** 360° **.

We can prove this using the angle sum of a triangle.

The rectangle above is split into two triangles by joining two vertices together across the diagonal.

As the sum of angles in a triangle is 180° , we can add two lots of 180° together, making the **angle sum of a quadrilateral **equal to ** 360° **.

This is the same for all types of quadrilaterals.

The four angles in any quadrilateral always add to 360° , but there are a few key properties of quadrilaterals that can help us calculate other angles.

**Parallelogram**:

- Opposite angles are the same (congruent)
- Opposite sides are the same
- Two pairs of supplementary angles (co-interior)
- Vertically opposite angles at the intersection of the diagonals

**Rectangle:**

- All the properties of a parallelogram
**and** - All edges meet at right angles

**Kite**:

- The diagonals are perpendicular lines
- One pair of opposite angles are congruent

**Rhombus**:

- All properties of a parallelogram
**and** - The diagonals 4 congruent triangles

**Square**:

- All the properties of a rectangle and a rhombus
**and** - The diagonals form 4 isosceles triangles

**Trapezium**:

- Angles at the intersection of the diagonals are
**vertically opposite.**

**Isosceles trapezium**:

- All properties of a trapezium
**and** - Two pairs of congruent angles

**Arrowhead**:

- Diagonals bisect at 90 degrees external to the shape
- One pair of congruent angles (if symmetrical)
- One obtuse angle

In order to find missing angles in a quadrilateral:

**Use angle properties to determine any interior angles.****Add all known interior angles.****Subtract the angle sum from 360° .****Form and solve the equation.**

Get your free angles in a quadrilateral worksheet of 20+ questions and answers. Includes reasoning and applied questions.

COMING SOONGet your free angles in a quadrilateral worksheet of 20+ questions and answers. Includes reasoning and applied questions.

COMING SOONABCD is a trapezium. Calculate the size of angle BCD , labelled x :

**Use angle properties to determine any interior angles**.

Angle fact:

The line AD is perpendicular to lines AB and CD so angle BAD = 90° .

2**Add all known interior angles**.

90+90+110=290^{\circ}

3**Subtract the angle sum from \pmb {360°} **.

Here, 360 - 290 = 70°

So, x = 70° .

Find the value of the missing angle x :

**Use angle properties to determine any interior angles**.

Angle fact:

- Angles in a quadrilateral add to equal 360° .

**Add all known interior angles**.

123+77+65=265^{\circ}

**Subtract the angle sum from \pmb {360°} **.

Here, 360 - 265 = 95°

So, x = 95° .

Calculate the missing angle for the following parallelogram:

**Use angle properties to determine any interior angles**.

Angle fact:

- Co-interior angles add to equal 180° .

**Add all known interior angles**.

x+32

**Form and solve the equation**.

Here,

Calculate the missing angle for the following parallelogram:

**Use angle properties to determine any interior angles**.

Angle fact:

- Diagonally opposite angles in a parallelogram are equal:

**Add all known interior angles**.

x+32+x+32=2x+64

**Form and solve the equation**.

Here,

Calculate the missing angle for the following quadrilateral

**Use angle properties to determine any interior angles**.

Angle fact:

- Angles on a straight line add to equal 180° :

**Add all known interior angles**.

99+105+75=279

**Subtract the angle sum from **\pmb {360°} .

Here, 360 - 279 = 81°

So, x = 81° .

By finding the value for x , calculate the value of each angle in the kite drawn below:

**Use angle properties to determine any interior angles**

Angle fact:

- One pair of diagonally opposite angles in a kite are the same size:

**Add all known interior angles**.

45+2x+x+2x=5=5x+45

**Form and solve the equation**.

Here,

As x = 63° we can find the value for the remaining angles in the kite by substituting the value onto each angle:

2x = 126°

So we have the four angles: 45°, 126°, 126°, and 63° .

We can check the solution by adding these angles together. They should add to equal 360° .

45 + 126 + 126 + 63 = 360° .

By finding the value for x , calculate the value of each angle in the quadrilateral:

**Use angle properties to determine any interior angles**.

For an irregular quadrilateral, there is only one angle property: the sum of the angles is equal to 360° .

**Add all known interior angles**.

100+2x-30+5x-40+3x=10x+110

**Form and solve the equation**.

Here,

**Mistaking the sum of angles in a quadrilateral with the angles in a triangle**

The angle sum is remembered incorrectly as 180° , rather than 360° . The sum of angles in a triangle is equal to 180° .

**Join all the diagonals**

When recalling the angle sum in a quadrilateral, students join all the diagonals together, creating 4 triangles. This makes their angle sum 720° which is also incorrect.

**Incorrect angle fact**

A common mistake is to use the incorrect angle fact or make an incorrect assumption to overcome a problem.

E.g.

Here the trapezium is assumed to be symmetrical (an isosceles trapezium) so the interior angles are easy to deduce. This is not always true and so you should use co-interior angles instead.

Here, the angle x should be equal to 105° and y should be equal to 60° due to co-interior angles in parallel lines.

1. ABCD is a rhombus. Given that CDA = 84^{\circ} calculate the value of a .

96^{\circ}

84^{\circ}

6^{\circ}

168^{\circ}

Diagonally opposite angles in a rhombus are equal.

Co-interior angles add to equal 180^{\circ} .

180-84=96^{\circ}

2. ABCD is a trapezium. Use the information below to calculate the value of b .

89^{\circ}

72^{\circ}

199^{\circ}

91^{\circ}

Co-interior angles add to equal 180^{\circ} .

180-89=91^{\circ}

3. ABCD is a parallelogram. Calculate the size of the angle BCD .

200^{\circ}

40^{\circ}

100^{\circ}

50^{\circ}

5x+4x=180 (co-interior)

9x=180\\
x=20\\
BCD=5x=100^{\circ} .

4. ABCD is a quadrilateral. Given that CE is a straight line, calculate the interior angle at D marked x .

279^{\circ}

155^{\circ}

223^{\circ}

137^{\circ}

Angles on a straight line add to equal 180^{\circ} .

Angles in a quadrilateral add up to 360^{\circ} .

5. ABCD is an isosceles trapezium. Calculate the value of y .

55^{\circ}

125^{\circ}

155^{\circ}

175^{\circ}

ADC=BCD

y=180-(3\times50-25)
y=180-125
y=55^{\circ}

6. ABCD is an irregular quadrilateral where BE is a straight line through C . Calculate the exact size of the angle y .

84^{\circ}

6^{\circ}

168^{\circ}

100^{\circ}

Angles on a straight line add to equal 180^{\circ} .

y=180-(140-2x)=2x+40\\
x+30+x+5x+20+2x+40=9x+90

Angles in a quadrilateral add to equal 360^{\circ} .

9x+90=360^{\circ}

As x=30^{\circ}, y=2x+40=230+40=100^{\circ} .

1. Show that the two quadrilaterals below are similar.

**(5 marks)**

Show answer

Co-interior angles add to equal 180^{\circ}

**(1)**

ABC=180-116=64180^{\circ} and ABC=EFG

**(1)**

FGH=180-64=116180^{\circ} and FGH=BCD

**(1)**

Diagonally opposite angles in a parallelogram are equal

**(1)**

All equal angles and sides are enlarged by a scale factor of 2

**(1)**

2.

(a) Calculate the size of angle \theta in the trapezium ABCD .

(b) What type of trapezium is ABCD ?

(c) State 2 properties about shape ABCD .

**(6 marks)**

Show answer

(a)

Angles on a straight line add to equal 180o and angle CDA=68^{\circ} .

**(1)**

Vertically opposite angles are equal and angle BCA=68^{\circ} .

**(1)**

DAB + CDA = 180^{\circ} because they are co-interior so \theta=112^{\circ}

**(1)**

(b)

An isosceles trapezium

**(1)**

(c)

Line AB and CD are parallel

**(1)**

One line of symmetry

**(1)**

3. Four matchsticks are dropped on the floor. They make a quadrilateral in the following arrangement

Use the information in the diagram to calculate the size of each interior angle of the shaded region.

**(5 marks)**

Show answer

Vertically opposite angles are equal

**(1)**

Angles on a straight line add to equal 180^{\circ}

**(1)**

Angles in a quadrilateral add to equal 360^{\circ} and 10x+90=360

**(1)**

**(1)**

Angles: 98^{\circ}, 95^{\circ}, 110^{\circ}, 57^{\circ}

**(1)**

You have now learned how to:

- Find unknown angles in any quadrilateral

- Interior angles of a polygon
- Exterior angles of a polygon
- Circle theorems

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