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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.

**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’.

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

** Angles on a straight line **

x+y=180^o

(The sum of angles on a straight line equals 180^o )

** Angles around a point **

e+f+g+h=360^o

(The sum of angles around a point

equals 360^o )

** Angles in a triangle **

A+B+C=180^o

(The sum of angles in a triangle equals 180^o )

**Vertically Opposite angles **

(Vertically opposite angles are the same

size)

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** 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.

Other examples of alternate angles:

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

**Step-by-step guide: **Alternate 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.

Other examples of corresponding angles:

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

** Step-by-step guide: **Corresponding angles

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

Other examples of co-interior angles:

i+j=180^{o}

k+l=180^{o}

m+n=180^{o}

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

**Step-by-step guide: **Co-interior angles

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.*

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

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

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

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

**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.**

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

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

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

\theta =180^o-50^o

\theta =130^o

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

**Highlight the angle(s) that you already kno**w.

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

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.**

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

\theta =60^o .

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

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

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

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

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

\theta =180-75

\theta =105^o .

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

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

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

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

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

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

Show that the two triangles are similar.

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

**Use a basic angle fact to calculate a missing angle.**

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 .

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

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.

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**.

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

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

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

**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.

1. Calculate the size of angle \theta

\theta = 100^{\circ}

\theta =80^{\circ}

\theta =138^{\circ}

\theta =42^{\circ}

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

We can then use angles on a straight line:

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

2. Calculate the size of angle \theta

Not to scale

\theta = 62^{\circ}

\theta =118^{\circ}

\theta =298^{\circ}

\theta =28^{\circ}

Using co-interior angles, we can calculate:

180-62=118^{\circ}

Then we can label the corresponding angle

118^{\circ}:

Since opposite angles are equal,

\theta=118^{\circ}

3. Calculate the angle \theta

Not to scale

\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}:

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

Not to scale

\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}

We can then use alternate angles to see that

\theta=23^{\circ}

5. Calculate the size of angle \theta

Not to scale

\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}:

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}.

Finally, using corresponding angles, we can see that:

\theta=151^{\circ}.

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

Not to scale

\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}:

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}

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

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.

(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)**

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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