What is the Measure of Angle BHC and Angle BGC? And Why?

As the property of angle in a circle:

angle BAC = arc BC = 116 deg

=> angle BHC = (1/2) angle BAC =(1/2) x 116 = 58 deg

angle HCG = (1/2) arc HF = (1/2) x 24 = 12 deg

B, H , G are on the same line, then:

angle BHC + angle CHG = 180 deg

=> angle CHG = 180 - angle BHC = 180 - 58 = 122 deg

Consider sum of three angles in triangle CHG, then:

angle CHG + angle HCG +angle HGC = 180 deg

=> angle HGC = 180 - angle CHG - angle HCG = 180 - 122 - 12 = 46 deg

=> angle BGC = angle HGC = 46 deg

Hope this helps!

:)

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step-by-step explanation: