1. The problem states that the triangle has three angles labeled as $7w$, $7w$, and $6w$. 2. We know the sum of the interior angles of any triangle is always $180^\circ$. 3. Therefore, we can write the equation: $$7w + 7w + 6w = 180$$ 4. Combine like terms: $$20w = 180$$ 5. Solve for $w$ by dividing both sides by 20: $$w = \frac{180}{20} = 9$$ 6. So, the value of $w$ is $9^\circ$.