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### High School Mathematics - 28.4 Theorems - Circles

 Theorem: Angle is a semi circle is a right angle. Given : Angle AOB is in semi circle. To prove: Angle ACB = 90o Proof: Angle ACB = 1/2 angle AOB (angle subtended at the centre is twice angle at circumference) Angle AOB = 180o (straight line) Hence ACB = 90o Theorem: Angles in the same segment of a circle are equal.  Directions: Solve the following
 Q 1: ABCD is a cyclic quadrilateral and BC = CD. Show that AC bisects �BAD Answer: Q 2: In the figure, PAQ and RAS are straight lines. Show that angPXR = angQYS. Answer: Q 3: If diagonals AC and BD of a quadrilateral intersect at M. Prove that a line drawn through M to bisect any side of the quadrilateral is perpendicular to the opposite side.Answer: Q 4: find x. 90 degrees110 degrees55 degrees Q 5: In the figure, CAD and CBE are straight lines. If CA is the diameter of the circle ABC, find angle ADE 45 degrees180 degrees90 degrees Q 6: Angles in a semicircle is always ____.270 degrees90 degrees180 degrees Q 7: Find x. 35 degrees140 degrees70 degrees Q 8: Find the angle x. 17.5 degrees70 degrees90 degrees Question 9: This question is available to subscribers only! Question 10: This question is available to subscribers only!

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