Cyclic Quadrilaterals

A cyclic quadrilateral is a quadrilateral with all four vertices or corners lying on the circumference of a circle.

Properties

  • The opposite angles add up to 180°(supplementary angles)

(abbreviated to opp. Y10_Cyclic_Quadrilaterals_01.gifs of cyclic quad are supp.)

Y10_Cyclic_Quadrilaterals_02.gif

a + c = 180o
b + d = 180°

 

 

  • The exterior angle of a cyclic quadrilateral is equal to the interior opposite angle.

(abbreviated to ext. Y10_Cyclic_Quadrilaterals_01.gifof cyclic quad = int.opp. Y10_Cyclic_Quadrilaterals_01.gif)

 

 

Y10_Cyclic_Quadrilaterals_03.gif

e = f

 
Examples
Answer

Find the value of the angles and give reasons.

Y10_Cyclic_Quadrilaterals_04.gif

x = 180° − 121° ( opp. Y10_Cyclic_Quadrilaterals_01.gifs of cyclic quad are supp.)
x = 59°

z = 75° ( ext. Y10_Cyclic_Quadrilaterals_01.gifof cyclic quad = int.opp. Y10_Cyclic_Quadrilaterals_01.gif)

 

y = 180° − 75° (adj. Y10_Cyclic_Quadrilaterals_01.gifs on st. line are supp.)
y = 105°