Geometry - Arcs created by Secant Lines

[marenubium]

Homework Statement

This picture:
http://i.imgur.com/n015WjU.png

It's drawn with exactly the amount of information from the worksheet. Specifically, the two secants meet at a point, with an angle of 28 degrees between them. Both secants partition off an arc of 120 degrees. The goal is to find the arc labeled x.

Homework Equations

Theorem (1): If an angle's vertex lies on a circle, then the angle is equal to half of the subtended arc.
Theorem (2): An angle is a central angle IFF the subtended arc is equal to the measure of the angle.

The Attempt at a Solution

Visual aid:
http://i.imgur.com/14Xoqfz.png

I labeled the points where the secants touch the circle A, B, C, D I then drew the purple lines.

By theorem 1, angles ADB, CBD are both 60 degrees.

Thus angle BED is 60 degrees since BED is a triangle, formed out of three straight lines.

Since AD is a straight line, AEB + BED = 180 degrees.

Thus AEB = 120 degrees.

This violates Theorem 2 since E is clearly not the center of the circle. I'm honestly not sure what I'm doing wrong. I've stared at this on and off for a couple hours now. Thanks for any directions.

 

[ehild]

Homework Statement

This picture:

It's drawn with exactly the amount of information from the worksheet. Specifically, the two secants meet at a point, with an angle of 28 degrees between them. Both secants partition off an arc of 120 degrees. The goal is to find the arc labeled x.

Homework Equations

Theorem (1): If an angle's vertex lies on a circle, then the angle is equal to half of the subtended arc.
Theorem (2): An angle is a central angle IFF the subtended arc is equal to the measure of the angle.

The Attempt at a Solution

Visual aid:
http://i.imgur.com/14Xoqfz.png

I labeled the points where the secants touch the circle A, B, C, D I then drew the purple lines.

By theorem 1, angles ADB, CBD are both 60 degrees.

Thus angle BED is 60 degrees since BED is a triangle, formed out of three straight lines.

Since AD is a straight line, AEB + BED = 180 degrees.

Thus AEB = 120 degrees.

This violates Theorem 2 since E is clearly not the center of the circle. I'm honestly not sure what I'm doing wrong. I've stared at this on and off for a couple hours now. Thanks for any directions.

Use the central angles. Theorem 2 is not true for angle AEB. What is subtended arc?

 

[ehild]

Some more hints:

Determine the green angles, then the red ones. With the 28° angle and the red ones, you get the blue angle, and then x.

 

[marenubium]

Thank you. I guess my problem is I forgot that I could manufacture central angles even though the figure didn't contain the center of the circle.