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### High School Mathematics - 28.5 Angle at Centre Theorem

 : Angle subtended by an arc at the centre is twice the angle subtended by the angle at the circumference. Given: ACB is the major arc of the circle. Angles AOB and ACB stand on the same arc. To prove: Angle AOB = 2 Angle ACB Proof: From triangle AOC, x = a+a (Exterior angle of a triangle) Hence x = 2a From triangle OBC, y = b+b Hence y = 2b Angle AOB = x + y Hence angAOB = 2(a+b) Hence Angle AOB = 2 Angle ACB Example: Find angle a. Angle at circumference = 1/2 angle at centre. Angle a = 70o Answer: 70o Directions: Solve the following
 Q 1: Answer: Q 2: Find angle ACB. 54o90o27o Q 3: If the sum of the angles at the circumference and the centre is 120 degrees. Find the angle at centre.70o80o40o Q 4: Consider triangle ABC in a circle, if one angle at the centre is 200 degrees, what is the other angle at the centre?100o160o120o Q 5: Answer: Q 6: If one angle at the circumference of a circle is 55 degrees , find the angle at centre.100o55o110o Q 7: Answer: Q 8: Consider triangle ABC inscribed in a circle, if ang ABC is 50 degrees and is in the major arc, what is the angle at the centre.50o100o75o Question 9: This question is available to subscribers only! Question 10: This question is available to subscribers only!

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