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iGCSE (2021 Edition)

28.02 Cyclic quadrilaterals (Extended)

Worksheet
Cyclic quadrilaterals
1

The diagram shows a cyclic quadrilateral. Solve for m.

2

In the following figure, ABCD is inscribed inside a circle:

a

What is the sum of any two opposite angles in the quadrilateral ABCD?

b

Can a parallelogram be inscribed inside a circle? Explain your answer.

c

Can a rectangle be inscribed inside a circle? Explain your answer.

3

Consider the following figure:

Determine the value of d.

4

Find the value of the pronumerals in the following diagrams. Give reasons for your answer.

a
b
c
d
5

Find the value of y in the following diagram:

6

Consider the following figure:

a

Find the value of x.

b

Find the value of y.

7

In the diagram, \angle AOC = 174 \degree, where O is the centre of the circle:

a

Determine the size of reflex \angle AOC.

b

Determine the size of \angle ABC.

c

Can a circle be drawn through the vertices of quadrilateral OABC? Explain your answer.

8

In the diagram, O is the centre of the circle:

a

Solve for the value of m, giving reasons for your answer.

b

Solve for the value of n, giving reasons for your answer.

9

In the diagram, O is the centre of the circle.

a

Solve for p, giving reasons for your answer.

b

Solve for q, giving reasons for your answer.

c

Solve for r, giving reasons for your answer.

10

Consider the given diagram:

a

Find the value of p, giving reasons for your answer.

b

Find the value of q, giving reasons for your answer.

11

In the diagram, \angle ADB = 32 \degree and \angle DBA = 43 \degree. Find the size of the angle marked m, giving reasons for your answer.

12

In the diagram, AB \parallel DC and \\ \angle BCE = 98 \degree . DC is produced to point E.

Find the size of \angle BAD.

13

In the diagram, \angle DBA = 27, \angle CAB = 49, \angle CBD = 21 and \angle ABD = m. Solve for m, giving reasons.

14

In the following figure, consider the two circles intersect at points B and E.

If \angle BCD=94\degree. Find \angle BAF.

Proofs
15

In the diagram, O is the centre of the circle. Show that x and y are supplementary angles.

16

In the diagram, ABCD is an isosceles trapezium, so AD = BC.

Prove that the points A, B, C and D are concyclic.

17

Consider the following diagram:

Prove that \angle BAC=126-z.

18

Consider the following diagram:

Prove AD || CF.

19

Consider the following diagram:

Prove that x = y.

20

Consider the figure:

a

Prove that \angle ABC=\angle CDE .

b

By proving two similar triangles, Prove that \angle BAD and \angle DCE are equal.

c

Hence prove that \\ EB \times EC = ED \times EA.

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Outcomes

0580E4.7I

Calculate unknown angles using angles in opposite segments are supplementary; cyclic quadrilaterals and the alternate segment theorem.

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