To find the value of $x$ in the given problem, let’s analyze the diagram.

The diagram shows:

• Two angles at point $D$, labeled as $(2x)^\circ$ and $(4x - 6)^\circ$.
• These two angles form a straight line, which means they are supplementary.

Since supplementary angles add up to $180^\circ$, we can set up the equation as follows:

$(2x) + (4x - 6) = 180$

Step-by-Step Solution:

1. Simplify the equation: $2x + 4x - 6 = 180$ $6x - 6 = 180$

2. Add 6 to both sides: $6x = 186$

3. Divide both sides by 6: $x = 31$

Answer:

$x = 31$

Would you like further clarification on this solution?

Here are some additional related questions to consider:

1. What are supplementary angles, and how are they identified?
2. How would you solve for $x$ if the angles were complementary instead of supplementary?
3. If another angle were added to this configuration, how would it affect the equation?
4. How can this method be applied to other geometry problems involving straight lines?
5. What are some other cases where angles around a point add up to specific values?

Tip: Always check if angles form a straight line or other familiar geometric patterns to quickly set up equations.