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GCSE Maths Geometry and Measure

Angles

Angles in parallel lines

Angles in Parallel Lines

Here we will learn about angles in parallel lines including how to recognise angles in parallel lines, use angle facts to find missing angles in parallel lines, and apply angles in parallel lines facts to solve algebraic 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 angles in parallel lines?

Angles in parallel lines are angles that are created when two parallel lines are intersected by another line called a transversal.

Angles in Parallel Lines Image 1

We can use the information given in the diagram to find any angle around the intersecting transversal.

To do this, we use three facts about angles in parallel lines:

Alternate angles, co-Interior angles, and corresponding angles.

  • Alternate angles are equal:

Angles in Parallel Lines Image 2

Sometimes called ‘Z angles’.

  • Corresponding angles are equal:

Angles in Parallel Lines Image 3

Sometimes called ‘F angles’

  • Co-interior angles add up to 180^o:

Angles in Parallel Lines Image 4

Sometimes called ‘C angles’.

What are angles in parallel lines?

What are angles in parallel lines?

Key angle facts

To explore angles in parallel lines we will need to use some key angle facts.

Angles on a straight line
 
Angles in Parallel Lines Key Facts Image 5
 
      x+y=180^o
 
(The sum of angles on a straight line equals 180^o )
 
 
 

      Angles around a point
 
        Angles in Parallel Lines Key Facts Image 6
 
         e+f+g+h=360^o
 
     (The sum of angles around a point
       equals 360^o )

Angles in a triangle
 
Angles in Parallel Lines Key Facts Image 7
 
    A+B+C=180^o
 
(The sum of angles in a triangle equals 180^o )

     Vertically Opposite angles
 
        Angles in Parallel Lines Key Facts Image 8
 
     (Vertically opposite angles are the same
      size)
 
 
 

Angles in parallel lines

We know that vertically opposite angles are equal and we can show this around a point within our parallel lines:

Angles in Parallel Lines Image 9

If we extend the transversal line so that it crosses more parallel lines, the angles that are made are maintained throughout the diagram for any line that is parallel to the original line AB.

Angles in Parallel Lines Image 10

Top Tip: for the same intersecting transversal, all the acute angles are the same size, and all the obtuse angles are the same size.

We group these angles into three separate types called alternate angles, co-interior angles and corresponding angles.

In this lesson, we will summarise each angle fact. However, you can look at each of them in detail by visiting their individual lesson pages.

Alternate angles

Alternate angles are angles that occur on opposite sides of the transversal line and have the same size.

Each pair of alternate angles around the transversal are equal to each other.
The two angles can either be alternate interior angles or alternate exterior angles.

Angles in Parallel Lines Alternate Angles Image 11

Other examples of alternate angles:

Angles in Parallel Lines Alternate Angles Examples Image 12

Angles in Parallel Lines Alternate Angles Examples Image 13

Angles in Parallel Lines Alternate Angles Examples Image 14

We can often spot interior alternate angles by drawing a Z shape:

Angles in Parallel Lines Alternate Angles Image 15

Step by step: Alternate Angles

Corresponding angles

The pairs of angles formed on the same side of the transversal that are either both obtuse or both acute and are called corresponding angles and are equal in size.

Each pair of corresponding angles on the same side of the intersecting transversal are equal to each other.

Angles in Parallel Lines Corresponding Angles Image 16

Other examples of corresponding angles:

Angles in Parallel Lines Corresponding Angles Examples Image 17

Angles in Parallel Lines Corresponding Angles Examples Image 18

Angles in Parallel Lines Corresponding Angles Examples Image 19

We can often spot interior corresponding angles by drawing an F shape:

Angles in Parallel Lines Corresponding Angles Examples Image 20

Step by step: Corresponding Angles (coming soon)

Co-Interior angles

Co-interior angles on the same side of an intersecting transversal add to 180^o .

Angles in Parallel Lines Co Interior Angles Examples Image 21

g+h= 180^o

Other examples of co-interior angles:

Angles in Parallel Lines Co Interior Angles Examples Image 22
  
i+j=180o

   Angles in Parallel Lines Co Interior Angles Examples Image 23
 
k+l=180o

   Angles in Parallel Lines Co Interior Angles Examples Image 24
 
m+n=180o

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

Angles in Parallel Lines Co Interior Angles Examples Image 25

Step by step: Co-interior angles (coming soon)

How to find a missing angle in parallel lines

In order to find a missing angle in parallel lines:

1 Highlight the angle(s) that you already know.

2 State the alternate angle, co-interior angle or corresponding angle fact to find a missing angle in the diagram.

3 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 find a missing angle in parallel lines

Explain how to find a missing angle in parallel lines

Angles in parallel lines worksheet

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

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Angles in parallel lines worksheet

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

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Angles in parallel lines examples

For each stage of the calculation we must clearly state any angle facts that we use.

Example 1: alternate angles

Calculate the size of the missing angle \theta . Justify your answer.

Angles in Parallel Lines Example 1
  1. Highlight the angle(s) that you already know.

Angles in Parallel Lines Example 1 Step 1

2State the alternate angle, co-interior angle or corresponding angle fact to find a missing angle in the diagram.

Angles in Parallel Lines Example 1 Step 2

Here we can label the alternate angle on the diagram as 50^o .

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

Angles in Parallel Lines Example 1 Step 3

Here as \theta is on a straight line with 50^o ,


\theta =180^o-50^o


\theta =130^o

Example 2: co-Interior angles

Calculate the size of the missing angle \theta . Justify your answer.

Angles in Parallel Lines Example 2

Angles in Parallel Lines Example 2 Step 1

Angles in Parallel Lines Example 2 Step 2


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

Angles in Parallel Lines Example 2 Step 3


We can see that as \theta is vertically opposite to 60^o ,


\theta =60^o .

Example 3: corresponding angles

Calculate the size of the missing angle \theta . Justify your answer.

Angles in Parallel Lines Example 3

Angles in Parallel Lines Example 3 Step 1

Angles in Parallel Lines Example 3 Step 2


Here we can label the corresponding angle on the diagram as 75^o .

Angles in Parallel Lines Example 3 Step 3


Here as \theta is on a straight line with 75^o ,


\theta =180-75


\theta =105^o .

Example 4: multiple-steps

Calculate the size of the missing angle \theta . Show all your working.

Angles in Parallel Lines Example 4

Angles in Parallel Lines Example 4 Step 1

Angles in Parallel Lines Example 4 Step 2


Opposite angles are equal so we can label the angle 110^o .

Angles in Parallel Lines Example 4 Step 3


Co-interior angles add up to 180^o . Here 180-110=70^o .

Angles in Parallel Lines Example 4 Step 4


θ is corresponding to 70+35 so θ = 70+35 = 105^o .

Example 5: similar triangles

Show that the two triangles are similar.

Angles in Parallel Lines Example 5

Angles in Parallel Lines Example 5 Step 1

Angles in Parallel Lines Example 5 Step 2.1

Angles in Parallel Lines Example 5 Step 2.2


Here, we can see that the two angles highlighted in green are on a straight line and so their sum is 180^o . This gives us the missing angle of 70^o .


We can also see there is a vertically opposite angle at the centre of the diagram. This is also 90^o .


Angles in Parallel Lines Example 5 Step 2.3


The smaller triangle now has a missing angle of 20^o as angles in a triangle add to equal 180^o .

Angles in Parallel Lines Example 5 Step 3.1


By stating the alternate angles from 70^o and 20^o we can see that θ=20^o and the other angle in the triangle is 70^o . The two triangles contain the same angles and are therefore similar.


Angles in Parallel Lines Example 5 Step 3.2

Example 6: Angles in parallel lines including algebra

Given that the sum of angles on a straight line is equal to 180^o , calculate the value of x . Hence or otherwise, calculate the size of angle 4x+30 .

Angles in Parallel Lines Example 6

Angles in Parallel Lines Example 6 Step 1

Angles in Parallel Lines Example 6 Step 2


Here we can state that 20^o is corresponding to the original angle.

Angles in Parallel Lines Example 6 Step 3

As the sum of angles on a straight line is 180^o, we have:.


4x+30+20=180^o


4x=130


x=32.5^o


Now that x=32.5^o,


4x+30=4(32.5)+30


=160^o

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 the alternate angles being on the alternate sides of the line.

  • 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.

Practice angles in parallel lines questions

1. Calculate the size of angle \theta

 

Angles in Parallel Line Practice Question 1

\theta = 100^{\circ}
GCSE Quiz False

\theta =80^{\circ}
GCSE Quiz False

\theta =138^{\circ}
GCSE Quiz True

\theta =42^{\circ}
GCSE Quiz False

Using corresponding angles, we can see the angle 42^{\circ}:

 

Angles in Parallel Line Practice Question 1 Answer

 

We can then use angles on a straight line:

\theta=180-42= 138^{\circ}

2. Calculate the size of angle \theta

 

Angles in Parallel Line Practice Question 2 Not to scale

 

 

\theta = 62^{\circ}
GCSE Quiz False

\theta =118^{\circ}
GCSE Quiz True

\theta =298^{\circ}
GCSE Quiz False

\theta =28^{\circ}
GCSE Quiz False

Using co-interior angles, we can calculate:

 

180-62=118^{\circ}

 

Angles in Parallel Line Practice Question 2 Answer 1

 

Then we can label the corresponding angle

118^{\circ}:

 

Angles in Parallel Line Practice Question 2 Answer 2

 

Since opposite angles are equal,

\theta=118^{\circ}

3. Calculate the angle  \theta

 

Angles in Parallel Line Practice Question 3Not to scale

 

 

\theta =21^{\circ}
GCSE Quiz False

\theta =159^{\circ}
GCSE Quiz False

\theta =69^{\circ}
GCSE Quiz False

\theta =79.5^{\circ}
GCSE Quiz True

Using opposite angles, we can see the angle 21^{\circ}.

 

Next we can label the alternate angle 21^{\circ}:

 

Angles in Parallel Line Practice Question 3 Answer

 

We can then use the fact that it is an isosceles triangle and so two base angles are equal:

 

\theta=\frac{180-21}{2}=79.5^{\circ}

4. Calculate the size of angle \theta

 

Angles in Parallel Line Practice Question 4 Not to scale

\theta =23^{\circ}
GCSE Quiz True

\theta =67^{\circ}
GCSE Quiz False

\theta =113^{\circ}
GCSE Quiz False

\theta =157^{\circ}
GCSE Quiz False

Using angles on a straight line, we can calculate:

180-(90+67)=23^{\circ}

 

Angles in Parallel Line Practice Question 4 Answer

 

We can then use alternate angles to see that

\theta=23^{\circ}

5. Calculate the size of angle \theta

 

Angles in Parallel Line Practice Question 5 Not to scale

\theta =151^{\circ}
GCSE Quiz True

\theta =88^{\circ}
GCSE Quiz False

\theta =121^{\circ}
GCSE Quiz False

\theta =59^{\circ}
GCSE Quiz False

Using angles on a straight line we can calculate the angles  92^{\circ} and 59^{\circ}:

 

Angles in Parallel Line Practice Question 5 Answer 1

 

Then the other angle in the triangle is:

180-(92+59)=29^{\circ}.

 

Using angles on a straight line we can calculate:

180-29=151^{\circ}.

 

Angles in Parallel Line Practice Question 5 Answer 1.2

 

Finally, using corresponding angles, we can see that:

\theta=151^{\circ}.

6. By calculating the value of x , find the value of \theta

 

Angles in Parallel Line Practice Question 6  Not to scale

\theta =150^{\circ}
GCSE Quiz False

\theta =90^{\circ}
GCSE Quiz False

\theta =30^{\circ}
GCSE Quiz False

\theta =85^{\circ}
GCSE Quiz True

30x-25 and 20x+5 are alternate angles. Therefore, we can write:

30x-25=20x+5.

 

We can then solve this to find x :

\begin{aligned} 30x-25&=20x+5\\ 10x-25&=5\\ 10x&=30\\ x&=3 \end{aligned}

 

Given that x=3 ,

30 \times 3-25=65

 

Using opposite angles, we can see that the angle inside the triangle is 65^{\circ}:

 

Angles in Parallel Line Practice Question 6 Answer

 

Using angles in a triangle, we can calculate the third angle in the triangle:

180-(65+30)=85^{\circ}.

 

Then using opposite angles,

\theta=85^{\circ}

Angles in parallel lines GCSE questions

1. (a) Below is a diagram showing two parallel lines intersected by a transversal:

 

Angles in Parallel Lines Exam Question 1

 

Write an equation connecting r and s.

 

(b) Given that the ratio of the angles r : s  is equivalent to 3 : 5 , write another equation connecting r and s.

 

(2 marks)

Show answer

(a)  r + s = 180

(1)

 

(b)  5r = 3s

(1)

2. Lines AB and CD are parallel.

 

Angles in Parallel Lines Exam Question 2

 

(a) By finding the value of x , calculate the exact value of z^{\circ} .

 

(b) Calculate the value of y^{\circ}.

 

(4 marks)

Show answer

(a)

 

5x – 10 = 4x – 2

(1)

 

x = 8^{\circ}

(1)

 

4 × 8  −  2 = z = 30^{\circ}

(1)

 

(b)

 

y = 180 – 30 = 150^{\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

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