find the numerical value of angle ABC + angle CDE also give reason for that answer

Question

ABCDE is a pentagon that lies on the circumference of the circle, AE is the diameter of the circle 

Answer 1

Take a look at the following drawing:

 

https://drive.google.com/file/d/0BwfqpqJa4_wia25uTFIzUlBCVDA/view?usp=sharing&resourcekey=0-Q6ViGGSXFLz7c1dpsO_zWA

 

angle ABC + angle CDE = α + β + γ + δ

 

Now, the sum of all the interior angles of a pentagon = 180(n-2) = 180(5-2) = 540^o

 

From the drawing: 2α + 2β + 2γ + 2δ = 540^o

therefore, α + β + γ + δ = 270^o

 

Answer 2

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°.