To solve this problem:

Step 1: Analyze the diagram

• The two angles $51^\circ$ and $(13x - 1)^\circ$ form a linear pair because they are adjacent and on the same line. This means their sum equals $180^\circ$.

Step 2: Set up the equation

$51 + (13x - 1) = 180$

Step 3: Simplify the equation

$51 - 1 + 13x = 180$ $50 + 13x = 180$

Step 4: Solve for $x$

$13x = 180 - 50$ $13x = 130$ $x = \frac{130}{13}$ $x = 10$

Final Answer:

$x = 10$

If you have further questions, feel free to ask! Here are some related questions to consider:

1. Why do the angles on a straight line add up to 180°?
2. What would the equation look like if the angle was $(13x + 5)^\circ$?
3. Can this approach be extended to angles forming around a full circle?
4. How does solving $13x + 50 = 180$ relate to the properties of linear pairs?
5. How can you verify the solution for $x$ once obtained?

Tip: Always double-check your angle relationships (e.g., complementary, supplementary, etc.) before setting up an equation!