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Recognise Alternate Angles

In this worksheet, students will learn how to recognise alternate angles and use the key fact, that alternate angles are equal, to find the value of unknown angles.

'Recognise Alternate Angles' worksheet

Key stage:  KS 4

GCSE Subjects:   Maths

GCSE Boards:   AQA, Eduqas, Pearson Edexcel, OCR

Curriculum topic:   Geometry and Measures, Basic Geometry

Curriculum subtopic:   Properties and Constructions, Angles

Difficulty level:  

Worksheet Overview

QUESTION 1 of 10

Image of a newspaper 

 

Welcome to this angle-themed activity - we have some bad news and some good news for you!

 

Bad news: There are various types of angles that you will need to be able to recognise quickly and confidently in diagrams.

Good news: Once you have made this identification accurately, there is often no further working out involved.

 

 

On this occasion, we are looking at alternate angles.

 

Once we know that we are observing alternate angles, we can apply the rule that they are the same size.

 

Fantastic!

All we need to do now is to be able to recognise them.

 

 

Review the two diagrams below showing alternate angles:

 

Diagrams of alternate angles

 

Here are the key scenarios to observe in relation to alternate angles:

 

1. They are found within a set of parallel lines. The parallel lines are usually marked with little arrows or dashes.

2. They are found either side of the transversal (i.e. the line cutting through the parallel lines.)

3. They are sometimes called Z angles because they make a Z-shape. (It is best to use their proper name of 'alternate angles' though, as this is how they will be referred to on exam papers.)

4. They are equal in size.

 

 

 

That is all there is to it - we just have to be able to spot them in diagrams now. 

 

Right then, let's put what we know into action. 

 

In this activity, we will learn how to recognise alternate angles and use the key fact, that alternate angles are equal, to find the value of unknown angles and solve related problems. 

Review the collection of diagrams below:

 

Collection of six different diagrams with a missing angle labelled x

 

Which diagrams show examples of alternate angles?

a)

b)

c)

d)

e)

f)

Now review this new diagram:

Diagram showing a number of different angles labelled with letters

Which angle is alternate to g?

a

b

c

d

Explore the angles shown on the diagram below:

Diagram showing a number of different angles labelled with letters

Which pairs of angles are alternate to each other?

h and b

g and d

h and c

g and a

Investigate the known and unknown angles on the diagram below:

Diagram showing a number of different angles labelled with letters

Then select the correct answers to complete the statements below. 

 

 abcd108°72°
The alternate angle to 72° is...
This angle measures...

Investigate the known and unknown angles on this diagram next:

Diagram showing a number of different angles labelled with letters

Then type a number to complete the statements below. 

 

 abcd108°72°
The alternate angle to 72° is...
This angle measures...

Investigate the known and unknown angles on this diagram now:

Diagram showing a number of different angles labelled with letters

Now type a letter then a number to complete the statements below. 

 abcd108°72°
The alternate angle to 72° is...
This angle measures...

Review the known and unknown angles on this new diagram:

 

Diagram showing a number of different angles labelled with letters

 

Then select the correct answers to complete the statements below. 

 69°111°
a° is alternate to...
b° is alternate to...

Explore the known and unknown angles on the diagram below:

 

Diagram showing a number of different angles labelled with letters

 

Then select the correct answers to complete the statements below. 

 

 107°73°
The value of m° is...
The value of j° is...

Observe the known and unknown angles on the diagram below:

 

Diagram showing a number of different angles labelled with letters

What is the value of angle x in this image?

 107°73°
The value of m° is...
The value of j° is...

Investigate the known and unknown angles on this final diagram now:

Diagram showing a number of different angles labelled with letters

What is the value of angle e?

 107°73°
The value of m° is...
The value of j° is...
  • Question 1

Review the collection of diagrams below:

 

Collection of six different diagrams with a missing angle labelled x

 

Which diagrams show examples of alternate angles?

CORRECT ANSWER
a)
e)
EDDIE SAYS
Did you spot these alternate angles quite quickly? We can use the key clue from the Introduction - can you see any Z-shapes with both angles in the corners labelled? The images which show a Z-shape with labelled corners are diagrams a) and e). The pair of angles in these diagrams are both within the parallel lines on opposite sides of the transversal. Review the Introduction now before you move on if any of these terms are unfamiliar or you found recognising these alternate angles to be tricky.
  • Question 2

Now review this new diagram:

Diagram showing a number of different angles labelled with letters

Which angle is alternate to g?

CORRECT ANSWER
d
EDDIE SAYS
Angle g is our starting point. We know that the alternate angle to g will be on the opposite side of the transversal and within the parallel lines. So the alternate angle could only be d. Can you see the z-shape which has angles g and d in its corners?
  • Question 3

Explore the angles shown on the diagram below:

Diagram showing a number of different angles labelled with letters

Which pairs of angles are alternate to each other?

CORRECT ANSWER
g and d
h and c
EDDIE SAYS
Sometimes these angles can be tricky to spot if there are several angles shown in a collection. Consider each pair one at a time, ignoring all other angles in each case. Remember we are looking for two angles which are inside the parallel lines and on opposite sides of the transversal, or in a Z-shape. The pairs which show this accurately are g and d and h and c.
  • Question 4

Investigate the known and unknown angles on the diagram below:

Diagram showing a number of different angles labelled with letters

Then select the correct answers to complete the statements below. 

 

CORRECT ANSWER
 abcd108°72°
The alternate angle to 72° is...
This angle measures...
EDDIE SAYS
Hopefully, these alternate or Z-angles are becoming easier to spot now. Which angle is inside the parallel lines and on opposite sides of the transversal from the 72° angle shown? It's angle c and it will have exactly the same value as its partner.
  • Question 5

Investigate the known and unknown angles on this diagram next:

Diagram showing a number of different angles labelled with letters

Then type a number to complete the statements below. 

 

CORRECT ANSWER
EDDIE SAYS
Uh oh... this looks a bit different, doesn't it? Just as we were getting comfortable! All we need to do is to look for the parallel lines and the transversal as usual; these features are all we need. Can you draw a Z in your mind, with the angle 48° in one of its corners? x is alternate to the angle shown as 48°, and we know that alternate angles are equal. It's pretty great when there is no working out involved, isn't it?!
  • Question 6

Investigate the known and unknown angles on this diagram now:

Diagram showing a number of different angles labelled with letters

Now type a letter then a number to complete the statements below. 

CORRECT ANSWER
EDDIE SAYS
Let's start at the 65°, then jump across the line, stay inside the parallel lines and, hey presto, we are there! So angle a is alternate and has the same value to the angle shown as 65°. Quite often, we will be asked to explain our answer, so the justification we can use is simply: "Alternate angles are always equal."
  • Question 7

Review the known and unknown angles on this new diagram:

 

Diagram showing a number of different angles labelled with letters

 

Then select the correct answers to complete the statements below. 

CORRECT ANSWER
 69°111°
a° is alternate to...
b° is alternate to...
EDDIE SAYS
A helpful way of thinking about 'alternate' angles, is that they are on the other side. 'Alternate' can also mean different, so that is how we can remember that they are on different sides of the transversal. That is where the difference ends, because once we find the correct angles, they are always the same. So angle a is alternate to 111° whilst angle b is alternate to 69°.
  • Question 8

Explore the known and unknown angles on the diagram below:

 

Diagram showing a number of different angles labelled with letters

 

Then select the correct answers to complete the statements below. 

 

CORRECT ANSWER
 107°73°
The value of m° is...
The value of j° is...
EDDIE SAYS
How is your confidence in spotting these alternate angle pairs developing? Can you draw Z-shapes in your mind, with the angles m° and j° in each of the corners? m° is alternate to 107°, and j° is alternate to 73°.
  • Question 9

Observe the known and unknown angles on the diagram below:

 

Diagram showing a number of different angles labelled with letters

What is the value of angle x in this image?

CORRECT ANSWER
EDDIE SAYS
Okay, we were warned that these alternate angles may not always be easy to spot. To be successful with this challenge, we need to recall some facts about an isosceles triangle too. If we extend the line at the top of the triangle, we can clearly see a set of parallel lines. We need to find one of the base angles of our isosceles triangle next: 180° - 28° = 152° Both base angles in an isosceles triangle are the same, so: 152° ÷ 2 = 76° If we put this angle into our triangle, we can see that the 76° is alternate to our unknown angle x - bingo!
  • Question 10

Investigate the known and unknown angles on this final diagram now:

Diagram showing a number of different angles labelled with letters

What is the value of angle e?

CORRECT ANSWER
EDDIE SAYS
We had to finish with a flourish - of course we did! This is not as challenging as it looks. We can see there are parallel lines at the top and bottom. If we follow the transversal between these parallel lines, we can see that angle b is alternate to angle e. We can also see that the triangle shown is isosceles, because of the markings on its side. We know that the base angles of an isosceles triangle are the same, so we can now find angle b: 180° - 40° = 140° 140° ÷ 2 = 70° Alternate angles are equal, so angle e must also be 70°. Phew, we got there in the end! Great work! You have learnt how to recognise alternate angles and use the key fact, that alternate angles are equal, to find the value of unknown angles.
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