Find the measure of ZAOE in the diagram below.
90
100
120
130

Explanation:

The three upper angles

all combine together to form a 180 degree straight angle. Let's add up the angle expressions and set the result equal to 180. That will help us solve for x.

(angle AOB) + (angle BOC) + (angle COD) = 180

(3x-28) + (90) + (66-x) = 180

2x+128 = 180

2x = 180-128 subtract 128 from both sides

2x = 52

x = 52/2 dividing both sides by 2

x = 26

Since x = 26, we can find the missing angles like so

Note how:

(angle AOB) + (angle BOC) + (angle COD) = 50+90+40 = 180

to help confirm we have the correct x value.

Afterward, we need to find angle BOD which is the combination of angles BOC and COD

angle BOD = (angle BOC) + (angle COD)

angle BOD = 90 + 40

angle BOD = 130

Angles BOD and AOE are vertical angles. It might help to erase ray OC so you can see the two vertical angles better. Vertical angles are always congruent, which means angle AOE is 130 degrees as well.