Angles in a quadrilateral

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Types 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 equations

This topic is relevant for:

GCSE Maths Geometry and Measure Angles Angles In Polygons

Angles In A Quadrilateral

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.

What are angles in a quadrilateral?

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.

What are angles in a quadrilateral?

Angle properties 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
All sides are equal in length
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.
One pair of parallel sides, therefore two pairs of supplementary angles (co-interior)

Isosceles trapezium:

All properties of a trapezium and
Two pairs of congruent angles

Arrowhead:

Diagonals bisect at $90$ degrees external to the shape (if symmetrical)
One pair of congruent angles (if symmetrical)
One reflex interior angle

How to find missing angles in a quadrilateral

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.

How to find missing angles in a quadrilateral

Angles in a quadrilateral worksheet

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

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Angles in a quadrilateral worksheet

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

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Related lessons on angles in polygons

Angles in a quadrilateral is part of our series of lessons to support revision on angles in polygons. You may find it helpful to start with the main angles in polygons lesson for a summary of what to expect, or use the step by step guides below for further detail on individual topics. Other lessons in this series include:

Angles in a quadrilateral examples

Example 1: trapezium

$ABCD$ 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 $BAD = 90°$ .

2Add all known interior angles.

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

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

Here, $360 - 290 = 70°$

So, $x = 70°$ .

We could have also found this angle using the fact that angle $ABC$ and angle $BCD$ are co-interior angles and, therefore, must add to $180°$ .

Example 2: irregular quadrilateral

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, $x = 95°$ .

Example 3: parallelogram with one interior angle (form and solve)

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,

Example 4: parallelogram with one interior angle (form and solve)

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,

Example 5: exterior angle given

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.

$99+105+75=279^{\circ}$

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

Here, $360 - 279 = 81°$

So, $x = 81°$ .

Example 6: forming and solving equations

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 $360°$ .

$45 + 126 + 126 + 63 = 360°$ .

Example 7: forming and solving equations

Find the value for $x$ , given the values 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,

Common misconceptions

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 $60°$ and $y$ should be equal to $105°$ due to co-interior angles in parallel lines.

Practice angles in a quadrilateral questions

96^{\circ}

84^{\circ}

6^{\circ}

168^{\circ}

Diagonally opposite angles in a rhombus are equal.

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

$180-84=96^{\circ}$

89^{\circ}

72^{\circ}

199^{\circ}

91^{\circ}

Co-interior angles add to equal $180^{\circ}$ .
$180-89=91^{\circ}$

200^{\circ}

40^{\circ}

100^{\circ}

50^{\circ}

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

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

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 $360^{\circ}$ .

55^{\circ}

125^{\circ}

155^{\circ}

175^{\circ}

$ADC=BCD$

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

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

Angles in quadrilaterals GCSE questions

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 angles correspond and the sides are enlarged by a scale factor of $2$

(1)

(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 $180^{\circ}$ 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)

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)

x=17^{\circ}

(1)

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

(1)

Learning checklist

You have now learned how to:

Find unknown angles in any quadrilateral

The next lessons are

Still stuck?

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