Central Angle : an angle with its vertex at the center of the circle Inscribed Angle : an angle whose vertex on a circle and whose side contain chord of.

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Central Angle : an angle with its vertex at the center of the circle Inscribed Angle : an angle whose vertex on a circle and whose side contain chord of the circle Inscribed and Central Angle Created by ﺠﻴﻄ for Mathlabsky.wordpress.com

a b c  AOB = Central Angle A C B O  ACB = Inscribed Angle  a +  b =  c Proof : x Theorem Inscribed and Central Angle

A B C O p p q q 2p 2q Let  ACO = p, and  BCO = q OC = OA (radius)  ACO =  CAO OC = OB (radius)  BCO =  CBO a b c D  AOD = 2p &  BOD = 2q &  ACB = p + q  AOB = 2p + 2q   AOB = 2  ACB  AOB = 2  ACB

A B C D O  AOB = 2  ACB  AOB = 2  ADB  ACB =  ADB “If two inscribed angles intercept the same arc, than the angle are congruent”  ACB =  ADB

A B C O  AOC =  AOC = 2  ABC  ABC = 90 0 “An angle inscribed in a semicercle is a right angle”  ABC = 90 0

A B C D O Let m  1=2x  m  2 = x and m  3 = (360 – 2x) “If a quadrilateral is inscribed in a circle, then its opposite angles are supplementary” m  3 = (360 – 2x)  m  4 = ½(360 – 2x)  m  4 = 180 – x m  2 + m  4 = x + (180 – x)  BAD +  BCD = 180  BAD +  BCD = 180

A B D E C  ABD =  ACD  BAC =  BDC  AEB =  CED   ABE   CDE C D EE A B 

A B O C D E ● Let  AOB = x and  COD = y   ACB = ½ x   DBC = ½ y ½ x ½ y B C E A   AEB = ½ x + ½ y  AEB = ½  AOB + ½  COD  AEB = ½  AOB + ½  COD y X

A B C D E O x y Let  AOB = x and  COD = y   ADB = ½ x   DBC = ½ y E B D A ½ y ½ x   AEB = ½ x - ½ y  AEB = ½  AOB - ½  COD