To find the value of $x$, let's analyze the geometry of the triangle shown in the image.

Step 1: Sum of angles in a triangle

The interior angles of a triangle always add up to $180^\circ$. In the triangle, the angles are labeled as:

• One angle: $6x$
• Two other angles: $9x$ each (as they are labeled at the base of the triangle).

Thus, the sum of the angles is: $6x + 9x + 9x = 180^\circ$

Step 2: Simplify the equation

Combine like terms: $24x = 180$

Step 3: Solve for $x$

Divide both sides of the equation by 24: $x = \frac{180}{24} = 7.5$

Final Answer:

$x = 7.5$

Let me know if you'd like further details or explanations!

Here are 5 related questions you might find interesting:

1. What are the specific measures of the angles in the triangle?
2. How would the problem change if one of the angles was not $6x$ but instead $5x$?
3. Can this triangle be classified as isosceles or scalene? Why?
4. How does this problem connect to properties of supplementary angles?
5. What other methods could be used to verify the solution?

Tip: Always double-check your equations and substitutions when solving geometry problems to avoid simple errors.