In the given diagram, we have straight lines $AB$ and $CD$ intersecting at point $O$. The angles involving $x$ and $2x$ need to be found based on the information provided.

Let's break down the problem step by step:

Observations:

1. Right angle at $EOD$: There is a $90^\circ$ angle at point $O$, i.e., $\angle EOD = 90^\circ$.
2. Vertical angles: Angles opposite each other at an intersection of two straight lines are equal.
3. Supplementary angles: The sum of angles on a straight line (e.g., at point $O$) is $180^\circ$.

Step 1: Angle Relationship at Point $O$

From the diagram, we see:

• $\angle EOF = 2x$
• $\angle EOC = 30^\circ$
• $\angle EOD = x$
• $\angle EOA = 90^\circ$ (Right angle)

Since $AB$ and $CD$ are straight lines, the angles formed at point $O$ must sum to $180^\circ$ on any straight line.

Step 2: Solve for $x$

The angles on the line $EF$ (at $O$) are supplementary, meaning:

$\angle EOD + \angle EOF = 90^\circ$

This gives us the equation: $x + 2x = 90^\circ$ $3x = 90^\circ$ $x = 30^\circ$

Step 3: Check the other angles

Now, let’s verify the other angle relationships:

• $2x = 2(30^\circ) = 60^\circ$
• The total of $\angle EOF (2x = 60^\circ)$ and $\angle EOC (30^\circ)$ is: $60^\circ + 30^\circ = 90^\circ$ which matches the straight-line property.

Conclusion:

The value of $x$ is $30^\circ$.

Would you like more details or clarification? Here are five related questions to further your understanding:

1. How do vertical angles help in solving problems like this one?
2. Can you explain how supplementary angles work in geometric configurations?
3. What is the significance of identifying right angles in such problems?
4. How would the solution change if one of the angles had been given differently?
5. Why do straight lines imply that the sum of angles at a point must be $180^\circ$?

Tip: Always remember to look for right angles and straight lines, as they provide crucial clues for solving angle problems.