Co-interior angles

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In order to access this I need to be confident with:

Angle rules Angles in a triangle Angles in polygons Solving equations

This topic is relevant for:

GCSE Maths Geometry and Measure Angles Angles In Parallel Lines

Co-Interior Angles

Here we will learn about co-interior angles in parallel lines, including how to recognise co-interior angles, and apply this understanding to solve problems.

There are also angles in parallel lines 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 co-interior angles?

Co-interior angles in parallel lines occur in between two parallel lines when they are intersected by a transversal. The two angles that occur on the same side of the transversal always add up to 180º.

Co-interior angles add up to 180º

Co interior angles image 1

$g + h = 180^{\circ}$

Co interior angles image 2

$i + j = 180^{\circ}$

Co interior angles Image 3

$k + l = 180^{\circ}$

Co interior angles image 5

$m + n = 180^{\circ}$

We can often spot co-interior angles by drawing a C shape.

The two interior angles are only equal when they are both 90º.
In all other cases we can work out one of the co-interior angles by subtracting the other from 180º.

What are co-interior angles?

How to calculate with co-interior angles

In order to find a missing angle in parallel lines:

Highlight the angle(s) that you already know.
Use co-interior angles to find a missing angle.
Use basic angle facts to calculate the missing angle.

Steps 2 and 3 may be done in either order and may need to be repeated. Step 3 may not always be required.

Explain how to calculate with co-interior angles

Co-interior angles worksheet

Get your free co-interior angles worksheet of 20+ questions and answers. Includes reasoning and applied questions.

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Co-interior angles worksheet

Get your free co-interior angles worksheet of 20+ questions and answers. Includes reasoning and applied questions.

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Angles in parallel lines – co-interior angles examples

Example 1: co-interior angles

Calculate the size of the missing angle θ. Justify your answer.

Highlight the angle(s) that you already know.

2Use co-interior angles to find a missing angle.

Here we can label the co-interior angle on the diagram as 60º as 120 + 60 = 180º.

3Use a basic angle fact to calculate the missing angle.

Here as θ is vertically opposite 60º,

θ = 60º.

Example 2: co-interior angles

Calculate the size of the missing angle θ. Justify your answer.

Highlight the angle(s) that you already know.

Show step

Here we can use either the 118º angle or the 83º angle. For this example we will use the 118º angle as this explores an alternative method not yet described.

Use co-interior angles to find a missing angle.

Show step

As co-interior angles have a sum of 180º, 180 − 118 = 180º , the missing angle highlighted is 62º.

Use a basic angle fact to calculate the missing angle.

Show step

Here we have a trapezium with the sum of interior angles as 360º. Using this fact, we can calculate the value of θ:

\[\begin{aligned} \theta&=360^{\circ}-\left(118^{\circ}+62^{\circ}+83^{\circ}\right) \\\\ \theta&=97^{\circ} \end{aligned}\]

Example 3: co-interior angles with algebra

Calculate the size of the missing angle θ. Justify your answer.

Highlight the angle(s) that you already know.

Show step

We need to find the value of x first.

Use co-interior angles to find a missing angle.

Show step

As 11x and 4x are co-interior, we can state that

\[\begin{aligned} 11x+4x&=180^{\circ} \\ 15x&=180^{\circ} \\ x&=12^{\circ} \end{aligned}\]

This gives us the two co-interior angles of 11x = 132º. and 4x = 48º.

Use a basic angle fact to calculate the missing angle.

Show step

The angles in the triangle are 48º and 90º, since the sum of angles in a triangle is 180º.

180 – 48 – 90 = 180º so the third angle in the triangle is 42º.

Since $\theta$ and $42^o$ are on a straight line,

\[\begin{aligned} \theta&=180-42 \\\\ \theta&=138^{\circ} \end{aligned}\]

Common misconceptions

Mixing up angle facts

There are a lot of angle facts and it is easy to mistake alternate angles with corresponding angles. To prevent this from occurring, think about co-interior angles being inside the C shape.

Using a protractor to measure an angle

Using a protractor to measure an angle. Most diagrams are not to scale and so using a protractor will not result in a correct answer unless it is a coincidence.

Co-interior angles is part of our series of lessons to support revision on angles in parallel lines. You may find it helpful to start with the main angles in parallel lines 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:

Practice co-interior angles questions

\theta =136^{\circ}

\theta =44^{\circ}

\theta =143^{\circ}

\theta =37^{\circ}

Co interior angles practice question 1 answer

$\theta$ and $44^{\circ}$ are co-interior angles. Therefore,

\theta = 180-44=136^{\circ}

\theta =125^{\circ}

\theta =55^{\circ}

\theta =70^{\circ}

\theta =62.5^{\circ}

Using co-interior angles, we can calculate

$180-125=55^{\circ}$
Co interior angles practice question 2 answer

Using co-interior angles again, we can see that

\theta=180-55=125^{\circ}

\theta =110^{\circ}

\theta =70^{\circ}

\theta =40^{\circ}

\theta =140^{\circ}

Using co-interior angles, we can calculate

$180-110=70^{\circ}$
Co interior angles practice question 3 answer

Since it is an isosceles triangle, the other angle at the bottom of the triangle is $70^{\circ}$ as well.

Then, using angles in a triangle,

\theta=180-(70+70)=40^{\circ}

\theta =90^{\circ}

\theta =95^{\circ}

\theta =85^{\circ}

\theta =83^{\circ}

$(7 + \theta)^{\circ}$ and $88^{\circ}$ are co-interior angles therefore

Co interior angles practice question 4 answer

\begin{aligned} (7+ \theta) +88&=180\\\\ \theta&=180-(88+7)\\\\ \theta&=85^{\circ} \end{aligned}

\theta =68^{\circ}

\theta =44^{\circ}

\theta =112^{\circ}

\theta =22^{\circ}

Since the triangle is an isosceles triangle, we know that the other angle at the top of the triangle is $68^{\circ}.$

$(\theta+68)^{\circ}$ and $68^{\circ}$ are co-interior angles therefore

Co interior angles practice question 5 answer

(\theta+68) + 68 = 100

\theta = 180-(68+68)

\theta=44\degree

 $42^{\circ}$  and  $38^{\circ}$

 $30^{\circ}$  and  $30^{\circ}$

 $80^{\circ}$  and  $100^{\circ}$

 $78^{\circ}$  and  $102^{\circ}$

$3x+12$ and $2x+18$ are co-interior angles, and so add up to $180\degree$ .

Therefore, we can write

$3x+12+2x+18=180$

$5x+30=180$

$5x=150$

$x=30$
so

$3x+12=3 \times 30 + 12=102\degree$

$2x+18=2 \times 30+18=78\degree$

Co-interior angles GCSE questions

Co interior angles exam question 1

(a) Find the size of angle $x$ .

(b) Find the size of angle $y$ .

(3 marks)

Show answer

(a)

$180-38=142^{\circ}$ as co-inteior angles add to $180^{\circ}$

$180-142=38^{\circ}$ as angles on a straight line add to $180^{\circ}$

(2)

(b)

$180 – {(110 + 38)}=32^{\circ}$

(1)

2. Work out the value of $x$ .

Co interior angles exam question 2

(3 marks)

Show answer

$5x – 7 + 3x + 11 = 180$

(1)

$8x + 4 = 180$

(1)

$x=22^{\circ}$

(1)

3. Are the lines $AB$ and $CD$ parallel? Give a reason for your answer.

Co interior angles exam question 3

(2 marks)

Show answer

No,

$95+87=182^{\circ}$

(1)

When two lines are parallel, the co-interior angles add up to $180^{\circ}$

(1)

Learning checklist:

You have now learned how to:

Apply the properties of angles at a point, angles at a point on a straight line, vertically opposite angles

Understand and use the relationship between parallel lines and alternate and corresponding angles

The next lessons are

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