This problem involves determining the value of $x$ based on the angles given in the image. Let’s analyze:

• The two angles near point $B$ are $(5x + 2)^\circ$ and $(4x - 2)^\circ$.
• The point $B$ lies on the line segment $\overrightarrow{AC}$, meaning these two angles together form a straight angle ($180^\circ$).

Step-by-Step Solution:

1. Since $\angle ABC = 180^\circ$: $(5x + 2) + (4x - 2) = 180$

2. Simplify the equation: $5x + 4x + 2 - 2 = 180$ $9x = 180$

3. Solve for $x$: $x = \frac{180}{9} = 20$

Verify:

Substitute $x = 20$ into the expressions for the angles:

• $(5x + 2) = 5(20) + 2 = 100 + 2 = 102^\circ$
• $(4x - 2) = 4(20) - 2 = 80 - 2 = 78^\circ$

Adding these angles: $102^\circ + 78^\circ = 180^\circ$ This confirms the solution is correct.

Final Answer:

$x = 20$

Do you need further details or have any questions?

Related Questions:

1. What is the definition of supplementary angles, and how does it apply here?
2. Can this setup be extended to find unknowns for non-linear configurations?
3. What are other geometric conditions that can be solved using straight-line properties?
4. How can you verify angle relationships when parallel lines and a transversal are involved?
5. How do linear equations in geometry connect to algebraic problem-solving?

Tip:

Always double-check your solution by verifying that the calculated angles sum to the expected total.