Question

I’m confused on this one

Answer 1

A very useful result called the inscribed angle theorem tells us that, given any inscribed angle on a circle (an angle on the inside edge of a circle) the central angle that subtends the same arc is twice the measure of the inscribed angle. I've included a visual example for this problem if any of that vocabulary is unfamiliar.

Here, the inscribed angle ∠C = 87° is attached to the arc DEB. According to the inscribed angle theorem, the central angle swept out by DEB should then be 87 x 2 = 174°. To find the measure of arc DE, we note that arc DEB is just arc DE + arc EB. We're given that arc EB measures 76°, so we subtract that from 174 to find that arc DE = 174 - 76 = 98°

Answer 2

Step One.

Focus on the 60 degree arc. The diagram is redrawn below. Everything begins with the center which has been labeled as 0. We are going to be working with central angles in order to find out what arc DE is. The central angle for the 60o arc is 60o which comes from the arc itself. The arc and its central angle are the same.

Step Two

Find out <DCO and and <CDO are. The triangle formed by DCO is isosceles because DO and CO are both radii. The angles opposite these two lines are therefore also equal. Call <DCO and CDO equal to x.

x + x + 60 = 180

2x + 60 = 180

2x = 180 - 60

2x = 120

x = 120/2 = 60. Amazingly enough, this triangle is equilateral!!!!

Step 3. You can now find <OCB and CBO (which are both equal. The key is in finding <OCB

Given: <DCB =87 degrees.

<DCB = <DCO + <OCB

87 = 60 + <OCB

<OCB = 87 - 60 = 27o

The triangle is isosceles because the radius = CO And BO. Let OCB = OBC = x

2x + <COB = 180o

2*27 + COB = 180

54 + <COB = 180

<COB = 180 - 54 = 126

If you understand everything up to this point, you have the problem solved.

Step Four

What you are trying to solve for is <DOE. You know 3 out of 4 of the central angles. If you can find the 4th one, you have the problem solves.

Given: The four central angles add up to 360 degrees.

<DOC = 60; <COB = 126; < BOE = 76; <DOE = ????

<DOE = 360 - (<COB + <DOC + <BOE)

<DOE = 360 - 76 - 60 - 126

<DOE = 360 - 262

<DOE = 98 Answer