The problem requires us to find the value of $k$ given that the sum of two angles, $(9k+6)^\circ$ and $(4k+5)^\circ$, forms a straight angle. A straight angle measures $180^\circ$.

1. Problem Description

The problem requires us to find the value of $k$ given that the sum of two angles, $(9k+6)^\circ$ and $(4k+5)^\circ$, forms a straight angle. A straight angle measures $180^\circ$.

2. Solution Steps

The sum of the two angles must equal 180 degrees. So we have:

$(9k+6) + (4k+5) = 180$

Combine like terms:

$13k + 11 = 180$

Subtract 11 from both sides:

$13k = 180 - 11$

$13k = 169$

Divide both sides by 13:

$k = \frac{169}{13}$

$k = 13$

3. Final Answer

$k = 13$