21C: 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 180o 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 = 360angles 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
Questions to do Ex 21C page 717 questions 1 2 3 4 5
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