1. 21C: Cyclic quadrilaterals

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

3. 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  )  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  )

4. 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   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 

5. 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   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 

6. Questions to do  Ex 21C page 717 questions 1 2 3 4 5