A "BIG" circle refers to a question that requires the use of all (or most) of your circle angle formulas in one problem. It may also be necessary to apply other strategies to find missing angles.

Solution: Start by finding the arcs, and then find the angles in any order that you wish.

 Find the measures of the arcs: 2x - 16 + x + 40 + x + 60 = 360 4x + 84 = 360 x = 69 Label the diagram with the arcs. ∠1 is an inscribed angle m∠1 = ½ arc = ½ (122) m∠1= 61º ∠2 is an inscribed angle m∠2 = ½ arc = ½ (60) m∠2= 30º ∠3 is an inscribed angle m∠3 = ½ arc = ½ (109) m∠3= 54.5º ∠4 is an inscribed angle m∠4 = ½ arc = ½ (122) m∠4= 61º ∠5 is an inscribed angle m∠5 = ½ arc = ½ (69) m∠4= 34.5º ∠6 is formed by a tangent and a chord m∠6 = ½ arc = ½ (60) m∠6= 30º ∠7 is "tricky"!!! No ∠ formula works! ∠7 & ∠4 linear pair m∠7 + m∠4 = 180º m∠7 + 61º = 180 m∠7 = 119º ∠8 is formed outside by tangent and secant m∠8 = ½ difference of arcs = ½ (122 - 60) m∠8= 31º ∠s 9, 10, 11, 12 formed by 2 intersecting chords m∠9 = ½ sum of arcs = ½ (109 + 60) m∠9 = 84.5º m∠10 = 95.5 (linear pair) m∠11 = 84.5º (vertical ∠) m∠12 = 95.5º (vertical ∠) ∠13 an inscribed angle m∠13 = ½ arc = ½ (69) m∠13= 34.5º ∠14 is an inscribed angle m∠14 = ½ arc = ½ (60) m∠14= 30º ∠15 is an inscribed angle m∠15 = ½ arc = ½ (109) m∠15= 54.5º As with all math problems, there are also other ways to arrive at these answers.