GCSE Maths Geometry and Measure

Angles

Corresponding Angles

# Corresponding Angles

Here is everything you need to know about corresponding angles for GCSE maths (Edexcel, AQA and OCR). You’ll learn how to recognise corresponding angles, and apply this understanding to solve problems.

Look out for the angles in parallel lines worksheets and exam questions at the end.

## What are corresponding angles?

Corresponding angles are when a transversal line crosses two parallel lines. The pairs of angles formed on the same side of the transversal that are either both obtuse or both acute and are equal in size.

Corresponding angles are equal

We can often spot corresponding angles by drawing an F shape.

### What are corresponding angles? ## How to calculate with corresponding angles

In order to calculate with corresponding angles:

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 a basic angle fact to calculate the missing angle.

### How to calculate with corresponding angles ## Corresponding angles examples

### Example 1: corresponding angles

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

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

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

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

Here θ is on a straight line with 75°

\begin{aligned} \theta&=180^{\circ}-75^{\circ} \\ \theta&=105^{\circ} \end{aligned}

### Example 2: corresponding angles

The new angles of 63° and 56° are corresponding angles to their original as shown above as they are on the same side of each transversal.

As the sum of angles in a triangle is 180°

\begin{aligned} \theta&=180^{\circ}-\left(63^{\circ}+56^{\circ}\right) \\ \theta&=61^{\circ} \end{aligned}

### Example 3: corresponding angles with algebra

Here we need to find the value for x to find the value of θ.

The relative position of the two angles 4x -14 and 3x+7 make them corresponding angles. This allows us to find the value of x as the two angles are congruent:

\begin{aligned} 4x-14&=3 x+7 \\ 4x&=3x+21 \\ x&=21^{\circ} \end{aligned}

As x = 21°, we have the two angles 4x – 14 = 3x + 7 = 70°.

Here we can combine the fact that we have an isosceles triangle and the sum of angles on a straight line is 180°

As the triangle is isosceles with the base of the triangle on one parallel line, the angles either side of the top vertex of the triangle are symmetrical (this can also be calculated using the alternate angle rule). These angles form a straight line. We can therefore find θ:

\begin{aligned} \theta&=180-(70+70) \\ \theta&=40^{\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 corresponding angles as being underneath the F 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.

### Practice corresponding angles questions

1. Calculate the size of angle \theta \theta=96^{\circ} \theta=84^{\circ} \theta=264^{\circ} \theta=74^{\circ} Using corresponding angles, we can see the angle 96^{\circ} . We can then use angles on a straight line: \theta=180-96=84^{\circ} . 2. Calculate the value of \theta . \theta=72^{\circ} \theta=36^{\circ} \theta=108^{\circ} \theta=144^{\circ} Using corresponding angles, we can see the angle \theta=72^{\circ} . We can then use angles on a straight line: \theta=180-72=108^{\circ} 3. Work out the size of angle \theta \theta=60^{\circ} \theta=58^{\circ} \theta=62^{\circ} \theta=122^{\circ} We start by calculating the missing angle in the triangle: 180-(62+60)=58 . We can then use corresponding angles to see that \theta=58^{\circ} . 4. Calculate the value for angle \theta \theta=106^{\circ} \theta=95^{\circ} \theta=115^{\circ} \theta=65^{\circ} Using corresponding angles, we can see that \theta is corresponding to 106+11 therefore \theta=106+11=115^{\circ} 5. Calculate the size of angle \theta \theta=12^{\circ} \theta=97^{\circ} \theta=109^{\circ} \theta=85^{\circ} Using corresponding angles, we can see the angle 97^{\circ} . Since vertically opposite angles are equal, the angle opposite is also 97^{\circ} . Using corresponding angles we can also see the angle 12^{\circ} . Then \theta=97-12=85^{\circ} . 6. By calculating the value for x , work out the size of each of the corresponding angles labelled in the diagram. 45^{\circ} \text{ and } 45^{\circ} 45^{\circ} \text{ and } 135^{\circ} 30^{\circ} \text{ and } 20^{\circ} 50^{\circ} \text{ and } 50^{\circ} 3x and 5x-30 are corresponding angles and so are equal. Therefore we can write 3x=5x-30 .
We can then solve this:
\begin{aligned} 3x&=5x-30\\ 0&=2x-30\\ 30&=2x\\ 15&=x \end{aligned}
\begin{array}{l} 3\times 15 = 45^{\circ}\\ 5\times 15-30 = 45^{\circ} \end{array}

### Corresponding Angles GCSE questions

1. (a) Work out the size of angle m. Give reasons for your answer.

(b) Work out the size of angle n.

(4 marks)

(a) m and 74 ^{\circ} corresponding

(1)

m=74^{\circ}

(1)

(b) Corresponding angles: (1)

180-74=106^{\circ}

(1)

2. (a) Work out the size of angle a.

(b) Work out the size of angle b.

(3 marks)

(a) a=37^{\circ} since they are corresponding

(1)

(b) 98^{\circ} is corresponding with b+37^{\circ}
so b+37=98

(1)

b=61^{\circ}

(1)

3. Work out the value of x . (3 marks)

Opposite angles equal (1)

2x+9=4x-3 since they are corresponding

(1)

\begin{aligned} 9&=2x-3\\ 12&=2x\\ 6&=x \end{aligned}

(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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