Angles in parallel lines

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Angle rules Angles in a triangle Angles in a quadrilateral Solving equations

This topic is relevant for:

GCSE Maths Geometry and Measure Angles

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.

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.

Properties of parallel lines

Alternate angles are equal:

Sometimes called $‘Z$ angles’.

Corresponding angles are equal:

Sometimes called $‘F$ angles’

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

Sometimes called $‘C$ angles’.

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:

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 $AB$ .

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.

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 guide: 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 guide: Corresponding angles

Co-Interior angles

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

g+h= 180^o

Other examples of co-interior angles:

Angles in Parallel Lines Co Interior Angles Examples Image 22

i+j=180^o

Angles in Parallel Lines Co Interior Angles Examples Image 23

k+l=180^o

Angles in Parallel Lines Co Interior Angles Examples Image 24

m+n=180^o

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 guide: Co-interior angles

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.

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.

DOWNLOAD FREE

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.

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.

Highlight the angle(s) that you already know.

Angles in Parallel Lines Example 2 Step 1

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

Angles in Parallel Lines Example 2 Step 2

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

Use a basic angle fact to calculate the missing angle.

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.

Highlight the angle(s) that you already know.

Angles in Parallel Lines Example 3 Step 1

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

Angles in Parallel Lines Example 3 Step 2

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

Use a basic angle fact to calculate the missing angle.

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.

Highlight the angle(s) that you already know.

Angles in Parallel Lines Example 4 Step 1

Use a basic angle fact to calculate the missing angle.

Angles in Parallel Lines Example 4 Step 2

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

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

Angles in Parallel Lines Example 4 Step 3

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

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

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.

Highlight the angle(s) that you already know.

Angles in Parallel Lines Example 5 Step 1

Use a basic angle fact to calculate a missing angle.

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 $180^o$ .

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

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 $4x+30$ .

Highlight the angle(s) that you already know.

Angles in Parallel Lines Example 6 Step 1

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

Angles in Parallel Lines Example 6 Step 2

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

Use a basic angle fact to calculate the missing 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

\theta = 100^{\circ}

\theta =80^{\circ}

\theta =138^{\circ}

\theta =42^{\circ}

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:

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

\theta = 62^{\circ}

\theta =118^{\circ}

\theta =298^{\circ}

\theta =28^{\circ}

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,

$\theta=118^{\circ}$

\theta =21^{\circ}

\theta =159^{\circ}

\theta =69^{\circ}

\theta =79.5^{\circ}

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:

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

\theta =23^{\circ}

\theta =67^{\circ}

\theta =113^{\circ}

\theta =157^{\circ}

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

$\theta=23^{\circ}$

\theta =151^{\circ}

\theta =88^{\circ}

\theta =121^{\circ}

\theta =59^{\circ}

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:

$\theta=151^{\circ}.$

\theta =150^{\circ}

\theta =90^{\circ}

\theta =30^{\circ}

\theta =85^{\circ}

$30x-25$ 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

The next lessons are

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