B is in the interior of angle AOC. C is in the interior of angle BOD. D is in the interior of angle COE. The measure of angle AOE is 162 degrees, COE is 68 degrees, and the measures of angles AOB, COD, and DOE are all equal. The question is to find the measure of angle DOA.

(it'd be a lot better if i'd post my sketch here, but anyway...)
I'm having a hard time what those three angles equal. If I knew what they equaled, I probably wouldn't have a hard time with the real question itself. I thought that angle AOC equaled 94 degrees just by measuring the angle with my protractor, but I think it's wrong because it may depend on how I draw the angles. My past results were that DOA equaled 123 degrees and 166. Can someone at least guide me to how you can find out what the measures of the three equal angles equal? Thanks much!

draw your figure.

if <cod=<doe, then each must be 1/2 of 68, or 34 deg. But <aob= the same measure, or 34 deg. So now, figure out what boc is.
<aob+<boc + <cod +<doe=162
then <boc= 162-3*34

check my thinking.
This problem has nothing to do with a protractor, the angles are in the mind.

Angle BOC equals 60 degrees, so again, AOC is 94 degrees. If I add AOB, BOC, and COD, I get the measure of DOA, which is 128. Thank you!!!

yes

To find the measure of angle DOA, we first need to determine the value of angles AOB, COD, and DOE.

Since angle AOB, COD, and DOE are all equal, let's denote their measure as x.

The sum of the angles in a triangle is 180 degrees. Therefore, the sum of angles AOE, COE, and angle O is:

162 + 68 + x = 180

Simplifying this equation, we have:

230 + x = 180

Subtracting 180 from both sides, we get:

x = -50

However, a negative angle is not possible in this scenario, so let's discard this solution.

Now, we have the measure of angle x, which is -50 degrees. Since we know that angle AOB, COD, and DOE are equal, we can write a similar equation for one of these angles.

Let's take angle AOB as an example:

AOB + COD + DOE = 180

Substituting x = -50:

AOB + (-50) + (-50) = 180

Simplifying:

AOB - 100 = 180

Adding 100 to both sides:

AOB = 280

Since angle AOC contains angles AOB and BOC, we can subtract angle AOB from angle AOC to find the measure of angle BOC:

AOC - AOB = BOC

Substituting the given angle measures:

(162 + 68) - 280 = BOC

Simplifying:

230 - 280 = BOC

-50 = BOC

Again, a negative angle is not possible, so let's discard this solution.

Now, let's consider the angles within angle DOA. Angle DOA is composed of angles COD, DOE, and angle COA:

DOA = COD + DOE + COA

Substituting the known angle measures:

DOA = -50 + (-50) + AOC

Given that angle COE is 68 degrees, angle COA can be obtained by subtracting angle COE from angle AOC:

COA = AOC - COE

Substituting the given angle measures:

COA = 162 - 68

COA = 94

Returning to the equation for DOA:

DOA = -50 + (-50) + AOC

Substituting COA:

DOA = -50 + (-50) + 94

DOA = -50

Once again, we have a negative angle. Thus, the solution is not valid.

Based on the given information and calculations, there seems to be an incongruity in the measurements provided. Please double-check the angle measures or provide any additional information if available.