31.3 Circle theorems

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31
.
3 Circle theorems


© The authors 2010. Edexcel GCSE Mathema
tics A Linear. This document may have been altered from the original

SPECIFICATION REFERE
NCES

GCSE 2010

GM j Understand and construct geometrical proofs using circle theorems

FS Process skills

Select the mathematical information to use

Use appropriate mathematical procedures

Choose appropriate language…to communicate result
s and solutions

FS Performance

Level 1 Select mathematics in an organised way to find solutions

RESOURCES

Resources

Pairs of compasses, rulers

Links

http://lgfl.skoool.co.uk/keystage4.aspx?id=317



Concepts and skills



Find missing angles on diagrams.



Prove

and use the facts that



the angle subtended by an arc at the centre of a circle is twice the angle subtended at any point
on the circumference



the angle in a semicircle is a right angle



the perpendicular from the centre of a circle to a chord bisects the c
hord.

Functional skills



L1 Construct geometric diagrams, models and shapes.

Prior key knowledge, skills and concepts

Students should already know how to state formally that



the angles in a triangle add up to 180°



the angles on a straight line add up to 18




the angles opposite the equal sides of an isosceles triangle are equal



the exterior angle of a triangle is equal to the sum of the two interior opposite angles.









31
.
3 Circle theorems


© The authors 2010. Edexcel GCSE Mathema
tics A Linear. This document may have been altered from the original


Starter



Put on the board some diagrams, with unknown angle values that can be foun
d using what students
have already learned about isosceles triangles in a circle and tangents to a circle, e.g.















Work out the missing angles and give a formal reason for your answers.
(Answers:

a

= 35º,
b

= 27º)

Main teaching and learning



Tell

students they are going to learn some circle theorems, explaining that they need to be able to
formally prove each theorem.



Go through the three theorems in detail and then
go through some exercises.

Common misconceptions



Tell students that when they draw

circles using their pair of compasses, it is best not to hold on to
the pencil end as this tends to make the end move, resulting in an inaccurately drawn circle.

Plenary



Give students copies of some problems involving circle theorems and supply a partial
explanation.
Ask them to suggest better wording.
31
.
4 More circle theorems


© The authors 2010. Edexcel GCSE Mathema
tics A Linear. This document may have been altered from the original

SPECIFICATION

GCSE 2010

GM j Understand and construct geometrical proofs using circle theorems

FS Process skills

Select the mathematical information to use

Use appropriate mathematical procedures

Choose app
ropriate language … to communicate results and solutions

FS Performance

Level 1 Select mathematics in an organised way to find solutions

Key words

cyclic quadrilateral, proof, prove

Resources

Pairs of compasses, rulers



Concepts and skills



Find missing a
ngles on diagrams.



Prove and use the facts that



angles in the same segment are equal



opposite angles of a cyclic quadrilateral sum to 180º



alternate segment theorem.

Functional skills



L1 Construct geometric diagrams, models and shapes.

Prior key knowledge,

skills and concepts

Students should already know how to state formally that



the angles in a triangle add up to 180°



the angles on a straight line add up to 180°



the angles opposite the equal sides of an isosceles triangle are equal



the exterior angle of a

triangle is equal to the sum of the two interior opposite angles.









31
.
4 More circle theorems


© The authors 2010. Edexcel GCSE Mathema
tics A Linear. This document may have been altered from the original


Starter



Put on the board some diagrams, with unknown angle values that can be found using the circle
theorems already covered.



Work out the missing angles and give a formal reason

for your answers.



e.g.










(
a

= 53º, because the

(
b

= 34º, because the angle

(
c

= 57º, because the angle in




perpendicular from the

at the centre of the circle

a semicircle is a right angle.)




centre of a circle to a

is twice the angle a
t the




chord bisects the chord.)

circumference.)

Main teaching and learning



Tell students they are going to learn some more circle theorems, explaining that they need to be
able to formally prove each theorem.



Go throu
gh the three theorems in detail
and

then
go through some exercises.

Plenary



Give students copies of some problems involving circle theorems and supply a partial explanation.
Ask them to suggest better wording.