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iGCSE (2021 Edition)

23.02 Cyclic quadrilaterals (Extended)

Lesson

A cyclic quadrilateral is a four-sided shape that has all its vertices touching the circle's circumference, such as the one shown below.

 

Cyclic quadrilaterals

The opposite angles in a cyclic quadrilateral add up to $$180°.

 

Proof:

 

  $$ABCD is a cyclic quadrilateral (given)
Join $$AC to $$BD  
$$CAB+ABC+ACB $$= $$180° (angle sum of a triangle)
$$CAB $$= $$CDB (angles in the same segment of a circle are equal)
$$ACB $$= $$ADB (angles in the same segment of a circle are equal)

Therefore, adding the previous two statements we get

$$ACB+CAB=ADB+CDB $$= $$ADC  
$$ACB+CAB+ABC $$= $$180° then - Adding $$ABC on both sides
$$ACB+CAB+ABC $$= $$180° (Angle sum of a triangle)
$$ADC+ABC $$= $$180°  
$$BAD+BCD=360°(ADC+ABC) $$= $$180°  

 

Remember!

The converse of this theorem is also true.

If a pair of opposite angles of a quadrilateral is supplementary, then the quadrilateral is cyclic.

 

Practice questions

Question 1

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

Question 2

Consider the figure:

  1. Prove that $$ABC = $$CDE.

  2. By proving two similar triangles, Prove that $$BAD and $$DCE are equal.

  3. Using this prove that $$EB×EC=ED×EA.

 

 

 

 

 

 

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