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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: 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 2: In the figure, CAD and CBE are straight lines. If CA is the diameter of the circle ABC, find angle ADE45 degrees90 degrees180 degrees Q 3: Two circles intersect each other at points A and B. If AP and BQ be the respective diameters, show that PBQ is a straight line.Answer: Q 4: Find x.70 degrees35 degrees140 degrees Question 5: This question is available to subscribers only! Question 6: This question is available to subscribers only!