GCSE Maths Geometry and Measure

Circles, Sectors and Arcs

Subtended

Subtended Angles

Here we will learn about subtended angles, including subtended angles of a line, curve and within polygons.

There are also subtended angles worksheets based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if you’re still stuck.

What is a subtended angle?

A subtended angle is the angle between two lines at a point. 

E.g.

The side AC subtends the angle θ from point B .

We can, therefore, describe a subtended angle as the angle made from a given point. 

For circle theorems, a subtended angle is an angle within a circle that is created by two chords meeting at a point on the circumference of a circle.

The diagrams below show the angle subtended by arc AC from point B for two different circles.

What is a subtended angle

What is a subtended angle

Important notation

How to find subtended angles

In order to find subtended angles:

  1. Use angle facts to determine missing angles.
  2. Use the appropriate circle theorem to find the subtended angle.

How to find subtended angles

How to find subtended angles

Subtended angles worksheet

Get your free subtended angles worksheet of 20+ questions and answers. Includes reasoning and applied questions.

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Subtended angles worksheet

Get your free subtended angles worksheet of 20+ questions and answers. Includes reasoning and applied questions.

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Subtended angles examples

Example 1: angles in the same segment

In the diagram, ∠BAC=70° , ∠ACB=40° . Find ∠ADC .

  1. Use angle facts to determine missing angles.

As angles in a triangle total 180° , angle ABC = 180 - (70+40) = 70° .

2Use the appropriate circle theorem to find the subtended angle.

As angles in the same segment are equal and AC is a chord shared by the points B and D within the same segment, the angle ADC = angle ABC = 70° .

Example 2: angles in the same segment

The diagram shows a circle with centre D and ∠CAD=42° . Find ∠ABC .

Triangle ACD is isosceles as AD and AC are radii. This means that angle ADC = 180 - (42 + 42) = 96° .

The angle at the centre is twice the angle at the circumference and so as the angles ABC and ADC are subtended by the same arc AC , angle ABC is half of angle ADC :

ABC = 96 ÷ 2 = 48° .

Example 3: angles in a semicircle

The diagram shows a circle with centre C and ∠ABD=27° . Find ∠CAD .

Triangle ABC is isosceles as AC and BC are radii. Angle BAC = 27° .

The angle at the circumference of a semicircle is equal to 90 degrees as the line BD is the diameter. This means that angle CAD = 90 - 27 = 63° .

Example 4: cyclic quadrilateral

In the diagram, ∠CBD=43° and ∠CDB=22° . Find ∠BAD .

As angles in a triangle total 180° , angle BCD = 180 - (22+43) = 115° .

Opposite angles in a cyclic quadrilateral total 180° , so angle BAD = 180 - 115 = 65° .

Example 5: alternate segment theorem

In the diagram, O is the centre, ∠ACB=58° and AD and BC are parallel. Find ∠CDF .

As lines AD and BC are parallel, we can state that angle CAD is alternate to angle BCA and so angle CAD = 58° .

The angle at A is subtended by the chord CD . The angle CDF is in the alternate segment to angle CAD so angle CDF = 58° .

Example 6: angles in the same segment

In the diagram, ∠ABD=80° and DCE=9° . Find ∠DCE .

For this problem, we need to use circle theorems to state missing angles.

Angle BAD = Angle BED = 90° as angles in a semicircle are 90° . This means that angle BDA = 180 - (90 + 80) = 10° .

This means that BDE = 33° and so DBE = 180 - (90+33) = 57° .

The angles at B and C are subtended by the chord DE . This means that they are in the same segment and so DCE = 57° .

Common misconceptions

  • Add to \pmb{90/180/360} degrees
    Make sure you know the other angle facts including:
    • Angles on a straight line total 180°.
    • Angles at a point total 360°.
    • Angles in a triangle total 180°.
    • Angles in a quadrilateral total 360°.
  • Halving and doubling
    By remembering the angle at the centre theorem incorrectly, the student could double the angle at the centre, or half the angle at the circumference.
    Top tip: look at the angles. The angle at the centre is always larger than the angle at the circumference (this isn’t so obvious when the angle at the circumference is in the opposite segment).

  • Angles are the same
    Make sure that you know when two angles are equal. Look out for isosceles triangles and the angles in the same segment.
  • A diameter or a chord?
    The angle at the circumference is assumed to be 90° when the associated chord does not intersect the centre of the circle and so the diagram does not show a semicircle.

Practice subtended angles questions

1. In the diagram, ∠BAD=83^{\circ} , ∠ABD=41^{\circ} . Find ∠ACB .

83^{\circ}
GCSE Quiz True

41^{\circ}
GCSE Quiz False

56^{\circ}
GCSE Quiz True

82^{\circ}
GCSE Quiz False

ADB = 180 – (83+41) = 56^{\circ} (Angles in a triangle total 180^{\circ}
 
Angle ADC = Angle ACB (Angles in the same segment are equal)
 
ACB = 56^{\circ}

2. In the diagram, D is the centre, ∠ABC=38^{\circ} . Find ∠ACD .
 

38^{\circ}
GCSE Quiz False

52^{\circ}
GCSE Quiz True

76^{\circ}
GCSE Quiz False

104^{\circ}
GCSE Quiz False

ADC = 38\times2 = 76^{\circ} (the angle at the centre is twice the angle at the circumference)
 
Triangle ACD is isosceles so angle ACD = (180 – 76)\div2 = 52^{\circ}

3. In the diagram, ∠ACB=53^{\circ} . Find ∠BAE .
 

26.5^{\circ}
GCSE Quiz False

37^{\circ}
GCSE Quiz False

53^{\circ}
GCSE Quiz False

90^{\circ}
GCSE Quiz True

BE is a diameter
 
BAE = 90^{\circ}

4. In the diagram, ∠BCD=88^{\circ}, ∠ADE=49^{\circ}, ∠AED=78^{\circ} . Find ∠BAE .
 

29^{\circ}
GCSE Quiz True

53^{\circ}
GCSE Quiz False

39^{\circ}
GCSE Quiz False

44^{\circ}
GCSE Quiz False

BAD = 180 – 88 = 92^{\circ} (opposite angles in a cyclic quadrilateral total 180^{\circ} )
 
DAE = 180 – (78+49) = 53^{\circ} (angles in a triangle total 180^{\circ} )
 
BAE = 92 – 53 = 29^{\circ}

5. In the diagram, with centre O , ∠ABE=58^{\circ} . Find ∠AOB .
 

29^{\circ}
GCSE Quiz False

32^{\circ}
GCSE Quiz False

58^{\circ}
GCSE Quiz False

116^{\circ}
GCSE Quiz True

Angle ACB = 58^{\circ} (Angle ACB is in the alternate segment to angle ABE )
 
Angle AOB = 58\times2 = 116^{\circ} (angle at the centre is twice the angle at the circumference)

6. In the diagram, with centre O , ∠BCD=111^{\circ} . Find ∠AOB .
 

55.5^{\circ}
GCSE Quiz False

132^{\circ}
GCSE Quiz True

120^{\circ}
GCSE Quiz False

69^{\circ}
GCSE Quiz False

Angle ADC = 90^{\circ} and ACD is an isosceles triangle so
angle ACD = 45^{\circ}
 
Angle BCO = 111-45 = 66^{\circ}
 
Angle AOB = 66\times2 = 132^{\circ} (angle at the centre is twice the angle at the circumference)

Subtended angles GCSE questions

1. Triangle ABC is inscribed inside a circle with centre O . AB = AC = 5cm and BC = 6cm . Calculate the size of the angle subtended at A from the chord BC .

 

(3 marks)

Show answer

\theta=\cos^{-1}\left(\frac{a^2+b^2-c^2}{2ab}\right)

(1)

\theta=\cos^{-1}\left(\frac{7}{25}\right)

(1)

\theta=73.7^{\circ}

(1)

2. The points A, B, C, and D are points on the circumference of the circle. Lines AC and BD intersect at the point E . Calculate the size of angle CBD .

 

(4 marks)

Show answer

CAD = 180 – (68+35) = 77^{\circ}

(1)

Angles in a triangle total 180^{\circ}

(1)

CBD = CAD = 77^{\circ}

(1)

Angles at A and B are subtended by the same chord CD and so they are equal

(1)

3. In the diagram, angle DAC = 30^{\circ} . The angle at D and the angle at B are subtended by the chord AC . Calculate the size of angle ABC .
 

 

(4 marks)

Show answer

DAC is an isosceles triangle

(1)

ADC = 180 – (30+30) = 120^{\circ}

(1)

ABC = 120 2 = 60^{\circ}

(1)

Angle at the centre is twice the angle at the circumference

(1)

Learning checklist

You have now learned how to:

  • Apply and prove the standard circle theorems concerning angles, radii, tangents and chords, and use them to prove related results

The next lessons are

  •  Circle theorems

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