21 C Cyclic quadrilaterals What is a cyclic

21 C: Cyclic quadrilaterals

What is a cyclic quadrilateral? A cyclic quadrilateral is a 4 sided shape that has all four corners on the circumference of a circle.

Rules for cyclic quadrilaterals Theorem: The opposite angles of a cyclic quadrilateral add to 180 o Theorem: The outside angle of a cyclic quadrilateral is equal to the opposite inside angle. Also, all the angles in a cyclic quadrilateral add to 360 (angles A + B + C + D = 360 )

Worked example We know that in a cyclic quadrilateral, opposite angles have to add to 180 degrees Therefore, x = 180 – 75 x = 105 And, y = 180 – 120 y = 60

Worked example We know the angle at point B has to be 80 , because it lies on a straight line with 100 and we know that straight lines have a total of 180 , so we can put that in to help us. Therefore, y must be 100 because it is opposite to the angle at point B in the cyclic quadrilateral, and opposite angles add to 180 You could also use the rule that the outside angle of a cyclic quadrilateral is equal to the opposite inside angle, and the outside angle opposite to y is 100 , so y = 100 The opposite inside angle to x is 50 , therefore using the rule that the outside angle of a cyclic quadrilateral is equal to the opposite inside angle, we know that x = 50 We could also figure this out because the angle at point C (next to x) must be 130 (as it is opposite to the angle at point A, so they need to add to 180 ), and if this angle is 130 , it means that x must be 50 because they are on a straight line together and have to add to 180 Check your answer by making sure all the internal angles of the quadrilateral add to 360 80

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