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Identify Vertically Opposite Angles

In this worksheet, students will use the key fact, that vertically opposite angles are equal, to find the value of unknown angles using number and algebra.

'Identify Vertically Opposite 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

Elephant and a mouse on a see saw

 

If you are the mouse on the see saw above, it makes sense that you wouldn't want to face the elephant opposite you.

You would want someone opposite you of the same size, to make things equal and moving the seesaw a lot easier!

 

There are lots of things in maths that are equal; a fact which often makes problems easier for us to solve as a result. 

 

 

When two lines intersect with each other (cross over), four angles are formed.

 

The pair of angles which lie opposite each other at the point of intersection are called vertically opposite angles.

Vertically opposite angles are equal as shown in this diagram:

Diagram of opposite angles

As an added extra, we can see that the angles on each of the straight lines, also add up 180°.

 

 

 

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

 

In this activity, we will use the key fact, that vertically opposite angles are equal, to find the value of unknown angles using number and algebra and solve problems involving vertically opposite angles. 

Consider this diagram:

Diagram of parallel lines

From the list below, choose the angles which are vertically opposite to each other.

b and f

a and c

h and f

d and h

Observe the diagram below:

Diagram of opposite angles

What is the value of angle x?

b and f

a and c

h and f

d and h

Investigate this new diagram:

Diagram of opposite angles

What is the value of angle x?

b and f

a and c

h and f

d and h

In each of the diagrams below, identify the value of angle x in each case.

a) Diagram of opposite angles

b) Diagram of opposite angles

c) Diagram of opposite angles

 105°98°42°
In a) x =
In b) x =
In c) x =

Observe the diagram below:

Diagram of opposite angles

What is the value of x?

125°

55°

35°

25°

Investigate this new diagram:

Diagram of opposite angles

Calculate the value of x in this case. 

125°

55°

35°

25°

Consider this more complicated diagram:

Diagram of opposite angles

Then complete the table below to express the correct values of angles x, y and z

 28°98°82°
Value of x =
Value of y =
Value of z =

What is the value of angle x in the diagram below?

Diagram of opposite angles

 28°98°82°
Value of x =
Value of y =
Value of z =

Your final challenge!

 

What is the value of angle x in the diagram below?

Diagram of parallel lines

134°

92°

64°

46°

Let's review this diagram again now, considering the new pairs of angles below:

Diagram of parallel lines

From the list below, choose the angles which are vertically opposite to each other.

b and d

a and e

g and b

g and e

  • Question 1

Consider this diagram:

Diagram of parallel lines

From the list below, choose the angles which are vertically opposite to each other.

CORRECT ANSWER
a and c
h and f
EDDIE SAYS
Wow, this looks complex, doesn't it?! Let's find the points of intersection first, then the pairs of opposite angles are easier to spot. Now consider each pair of angles provided in turn. Let's ask ourselves, are the angles opposite each other and do they look the same size (approximately)? Also, you will notice that on each straight line one angle is acute and the other obtuse. So we simply need to pick the pairs that are either both acute or obtuse.
  • Question 2

Observe the diagram below:

Diagram of opposite angles

What is the value of angle x?

CORRECT ANSWER
EDDIE SAYS
Is angle x vertically opposite to the other angle provided? It is! We know the rule that: The pair of angles which lie opposite each other at the point of intersection are called vertically opposite angles, and vertically opposite angles are equal. So the value of angle x must be the same as its opposite - which is 80°.
  • Question 3

Investigate this new diagram:

Diagram of opposite angles

What is the value of angle x?

CORRECT ANSWER
EDDIE SAYS
Do you know the expression that opposites attract? They certainly do here! So x is also 92° - it's as simple as that.
  • Question 4

In each of the diagrams below, identify the value of angle x in each case.

a) Diagram of opposite angles

b) Diagram of opposite angles

c) Diagram of opposite angles

CORRECT ANSWER
 105°98°42°
In a) x =
In b) x =
In c) x =
EDDIE SAYS
In each diagram, we need to decide which angle the angle x is vertically opposite to. Did you locate each value accurately in the table to summarise your observations?
  • Question 5

Observe the diagram below:

Diagram of opposite angles

What is the value of x?

CORRECT ANSWER
25°
EDDIE SAYS
In this case, the angle 5x is an example of algebra - it sneaks in everywhere, doesn't it?! We can see that 5x is vertically opposite to the angle 128°. We can think of this as '5 lots of x is equal to 125°'. So to find the value of x on its own, we need to divide by 5: 125° ÷ 5 = 25°
  • Question 6

Investigate this new diagram:

Diagram of opposite angles

Calculate the value of x in this case. 

CORRECT ANSWER
EDDIE SAYS
Throw in a little bit of algebra again, not a problem for us! We can see that 3x is vertically opposite to the angle 36°. We can think of this as '3 lots of x is equal to 36°'. So to find the value of x on its own, we need to divide by 3: 36° ÷ 3 = 12°
  • Question 7

Consider this more complicated diagram:

Diagram of opposite angles

Then complete the table below to express the correct values of angles x, y and z

CORRECT ANSWER
 28°98°82°
Value of x =
Value of y =
Value of z =
EDDIE SAYS
We need to work out which angles are vertically opposite in this diagram. We can see that y is vertically opposite to 28°, so these two angles must have the same value. A straight line can be created starting at angle y and ending at 54°, which includes the missing angle x. We know that there is 180° in a straight line and we now know the value of y, so: x = 180° - 54° - 28° = 98° We can see that z is vertically opposite to x, so these two angles must also have the same value.
  • Question 8

What is the value of angle x in the diagram below?

Diagram of opposite angles

CORRECT ANSWER
EDDIE SAYS
Quite often we will have to find angles that are we not asked for directly in order to solve a problem. In this question, we need find the angle which is vertically opposite to x in order to find the value of x. We know that angles on a straight line add up to 180°, and we can draw a straight line from the angle 42° to the angle 87°, so: 180° - 42° - 87° = 51° We can see that x is vertically opposite to this angle, so these two angles must also have the same value.
  • Question 9

Your final challenge!

 

What is the value of angle x in the diagram below?

Diagram of parallel lines

CORRECT ANSWER
46°
EDDIE SAYS
This was a tricky one! The opposite angle to 46° will have the same value, so will be 46° too. Corresponding angles, which means the angles which occupy the same position at each intersection where a straight line crosses two others, are also equal. This second 46° angle and the angle x are corresponding angles, so will also be the same. Don't worry if this was tricky, you will learn more about corresponding angles in other activities. Great job! You can now use the key fact, that vertically opposite angles are equal, to find the value of unknown angles using both numbers and algebra.
  • Question 10

Let's review this diagram again now, considering the new pairs of angles below:

Diagram of parallel lines

From the list below, choose the angles which are vertically opposite to each other.

CORRECT ANSWER
b and d
g and e
EDDIE SAYS
Let's find the points of intersection again. Now consider each pair of angles provided in turn. We need to pick the pairs that are either both acute or obtuse.
---- OR ----

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