ANGLE PROPERTIES OF CIRCLES
TIP: When doing these sort of problems, remember:
1) Make sure you know basic geometrical facts such as "the angles in a triangle add up to 180o" etc. Many of the questions rely on knowledge such as this to fully or partially solve the problem.
2) Take a look at the tutorial Round Trips to remind you of the circle theorems.
3) Remember
to explain why you have answered in the way you have - relate each answer
to the theorem or property you used to work out the answer. Even if it seems
obvious and you are just writing down an answer you MUST indicate your method.
e.g. Angle ACB = 90o (angle in semicircle).
4) These sort of problems can look very tricky, however if you know the properties of angles and tangents and circles, and solve LOTS of circle problems, you'll find it will gradually sink in.
5) Don't make the mistake of learning the formal words of each theory off by heart, you won't understand it properly and I've not seen an exam question yet where you're asked to "state circle theory XXX" - just remember the titles e.g. "alternate segment", "angles in the same segment", "angle in semicircle" etc.. Practice doing the problems and think about why the theories have come about - it's all about seeing positions and relationships between curved and straight lines.
Solve the following:
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1. A circle is drawn with centre O, where A, B, C and D are points on its circumference. Find in terms of x, the following: a) angle ADB b) angle AOD c) angle ABD |
2. 
A, B, C and D are points on a circle a) Find the angle ACB b) Find the angle AXB and describe what this tells you about the point X. |
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3. CP and CQ are tangents to a circle which has centre O. a) Explain why the triangles CQO and CPO are congruent. b) If the angle PCQ = 72o, what is the reflex angle POQ ?
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4. ABCD are points on the circumference of the circle. a) Find the angle ABC b) Find the angle ACB
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5. a) Calculate the angle between the tangent and the line AC. b) Calculate the angle ABC.
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6. Line PQ is the tangent to the circle at point A a) Determine the angle QAC in terms of x. b) Calculate the value of x.
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7. AB and BC are parallel lines between the two straight lines ABX and DCX. a) Work out the following angles: b) Show that the triangle EBA is isosceles.
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8. ABC and ADE are straight lines. The diameter of the larger circle is CE. Express in terms of x, the following angles: a) ABD b) DBE c) BAD Explain why the statement BE x AC = AE x CD is true.
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