Calculating angles with circles

Calculating angles with circles

Pencil and paperOnline interactive
Overview
Using this Resource
Connecting to the Curriculum
Marking Student Responses
Working with Students
Further Resources
This task is about the angle between a tangent and a radius, and the sum of angles in a triangle and the sum of angles in a quadrilateral. 
Complete the following sentences by finding the size of the marked angles.  
In each case, justify your answer.  
The centre of each circle is marked with a dot.

Question 1Change answer

a)
Calculating-tangent-circle-diagram-i.png [Not drawn to scale]
Angle t =  ° because ...
   
Angle u =  ° because ...

Question 1Change answer

b)
diagram of a angles and a circle [Not drawn to scale]
Angles y + z =  °
Angle x =  ° because ...
Task administration: 
This task can be completed with pencil and paper or online (with some auto-marking).
Level:
5
Curriculum info: 
Maths, Geometry and Measurement, Shape
Keywords: 
angles, right angles, circles, tangents, angle properties
Description of task: 
Students calculate the size of marked angles using their knowledge of angle properties: the angle between a tangent and a radius, the sum of angles in a triangle and the sum of angles in a quadrilateral.
Learning Progression Frameworks
This resource can provide evidence of learning associated with
Geometric thinking, set 6
within the Mathematics Learning Progressions Frameworks.
Read more about the Learning Progressions Frameworks.
Answers/responses: 

 

Y11 (09/2001)

a)

i)
ii)

90 and because the angle between a tangent and a radius equals 90°. (Do not accept because it is a right angle.)
43 (Accept 180° - 47° - answer to a)i)) and because the sum of the angles in a triangle equals 180°.

 moderate
easy

b)

i)
ii)

 

180
132 and

  • because the sum of the angles in a quadrilateral equals 360° (Also accept 360° - 48° - answer to c) i)), or
  • because the sum of the angles in a triangle equals 180° (if the quadrilateral has been divided into 2 triangles through x and 48°).
  • because opposite angles in a cyclic quadrilateral sum to 180°.
 difficult
difficult

 

 

Diagnostic and formative information: 
  

Common error

Likely miscalculation

Likely reason

b)ii)

96

48 × 2

Applies rule of "angle at the centre is twice the angle at the circumference" to an inappropriate situation.