A challenge for all the genius mathematicians. Solve question (b) to gain a lot of points.
And quick..
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Go ahead and draw this situation.
First, draw AE, the diameter of the circle. Make it a horizontal line. Now, along one half of the circle starting at A and ending at E, mark points B, C, and D. Make sure the points are in that order (A, B, C, D, E). Remember that, since AE is a diameter, the arc AE that doesn't include B, C, and D is 180°.
The arc that does include B, C, and D is also 180°.
That means that AB + BC + CD + DE = 180°.
From now on, when I refer to arc AE, I'm referring to the semi-circle not including B, C, and D.
<ABC is an inscribed angle. It's intercepted arc is CD + DE + AE = CD + DE + 180°.
That means the angle's measure is half that: m<ABC = (1/2)(CD + DE + 180) <CDE is another inscribed angle, intercepting the arc BC + AB + AE = BC + AB + 180°.
That means its measure is half that: m <CDE = (1/2)(BC + AB + 180) = (1/2)(AB + BC + 180)
The sum of the angles is: <ABC + <CDE = (1/2)(CD + DE + 180) + (1/2)(AB + BC + 180)<ABC + <CDE = (1/2)(CD + DE + 180 + AB + BC + 180)<ABC + <CDE = (1/2)(AB + BC + CD + DE + 360) Remember, AB + BC + CD + DE = 180°, since it describes a semi-circle.
So: <ABC + <CDE = (1/2)(180 + 360)<ABC + <CDE = (1/2)(540)<ABC + <CDE = 270
So, the sum of the two angle measures is 270°.
First, draw AE, the diameter of the circle. Make it a horizontal line. Now, along one half of the circle starting at A and ending at E, mark points B, C, and D. Make sure the points are in that order (A, B, C, D, E). Remember that, since AE is a diameter, the arc AE that doesn't include B, C, and D is 180°.
The arc that does include B, C, and D is also 180°.
That means that AB + BC + CD + DE = 180°.
From now on, when I refer to arc AE, I'm referring to the semi-circle not including B, C, and D.
<ABC is an inscribed angle. It's intercepted arc is CD + DE + AE = CD + DE + 180°.
That means the angle's measure is half that: m<ABC = (1/2)(CD + DE + 180) <CDE is another inscribed angle, intercepting the arc BC + AB + AE = BC + AB + 180°.
That means its measure is half that: m <CDE = (1/2)(BC + AB + 180) = (1/2)(AB + BC + 180)
The sum of the angles is: <ABC + <CDE = (1/2)(CD + DE + 180) + (1/2)(AB + BC + 180)<ABC + <CDE = (1/2)(CD + DE + 180 + AB + BC + 180)<ABC + <CDE = (1/2)(AB + BC + CD + DE + 360) Remember, AB + BC + CD + DE = 180°, since it describes a semi-circle.
So: <ABC + <CDE = (1/2)(180 + 360)<ABC + <CDE = (1/2)(540)<ABC + <CDE = 270
So, the sum of the two angle measures is 270°.
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Answer:
The man up there is correct
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