The problem involves finding the measure of angle 1 in a triangle with the given angle measures: $79^\circ$, $20x + 7^\circ$, and $8x^\circ$.

Since the sum of the interior angles of a triangle is always $180^\circ$, we can set up the following equation: $20x + 7 + 8x + 79 = 180$

Let's solve this equation to find the value of $x$ and subsequently, the measure of angle 1.The solution for $x$ is $\frac{47}{14} \approx 3.36$.

Now, we need to calculate the measure of angle 1, which is given by the expression $20x + 7^\circ$. Let's plug in the value of $x$ to find the measure of angle 1.The measure of angle 1 is approximately $74.14^\circ$.

Would you like to see more details or do you have any other questions? Here are some related topics you might find interesting:

1. How does the sum of angles in a triangle relate to its type (acute, obtuse, right)?
2. What would change if one of the angles were variable instead of fixed?
3. How do we solve problems involving exterior angles of a triangle?
4. Can you explain how algebraic expressions are used to represent angles in geometry?
5. How to find the values of other angles in a triangle if more angles are expressed in terms of variables?

Tip: When solving for angles in triangles involving algebraic expressions, always ensure your algebraic manipulations are accurate by double-checking your work.