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### High School Mathematics - 28.6 Circles - Theorems

 Statement: If two arcs are congruent, then their chords are equal. Given: Arc ASB = Arc CTD To prove: AB = CD Proof: OA = OlC, OB = OlD (radii) Angle AOB = Angle COlD (given that the arcs are congruent) Hence by SAS Postulate on congruence AOB is congruent to COlD Hence AB = CD Statement: There is only one circle passing through three non collinear points. Example: In the given figure prove that AB = CD Solution: Arc BAD congruent Arc CDA Hence Arc BAD - Arc AD = Arc CAD - Arc AD Arc BA is congruent Arc CD Hence AB = CD Directions: Solve the following.
 Q 1: BC is a chord of circle with centre O. A is a point on the major arc BC as shown, prove that angBAC + angOBC = 90.Answer: Q 2: Find the missing angle.55o30o35o Q 3: Two diameters of a circle intersect each other at right angles. Prove that the quadrilateral formed by joining their end points is a square.Answer: Q 4: Two circles intersect each other at points A and B, if AP and AQ be the respective diameters of the circle prove that PBQ is a line.Answer: Q 5: AC and BD are chords of a circle that bisect each other. Prove that AC and BD are diameters.Answer: Q 6: D is the midpoint of side BC of isosceles triangle ABC with AB = AC. Prove that the circle drawn witrh either of the equal sides as diameter pass through D.Answer: Q 7: In a circle containing the equal chords AB and CD which are on the opposite side of the centre, E. Prove that BE = DE and AE = CE where E is the point of intersection of AD and BC.Answer: Q 8: Find angle x55o35o90o Question 9: This question is available to subscribers only! Question 10: This question is available to subscribers only!