To solve this problem, let's start by defining the measures of the angles:

Let:

• $m\angle 6 = x$
• $m\angle 7 = y$

The given information tells us that:

1. Angles $\angle 6$ and $\angle 7$ form a linear pair. Since they are a linear pair, their measures add up to 180 degrees: $x + y = 180$

2. Twice the measure of $\angle 6$ is twelve more than four times the measure of $\angle 7$: $2x = 4y + 12$

Now, we have two equations:

1. $x + y = 180$
2. $2x = 4y + 12$

Let's solve these equations step-by-step:

Step 1: Solve for $x$ in terms of $y$

From equation (1): $x = 180 - y$

Step 2: Substitute $x$ in the second equation

Substitute the expression for $x$ into equation (2): $2(180 - y) = 4y + 12$

Simplify and solve for $y$: $360 - 2y = 4y + 12 \implies 360 - 12 = 4y + 2y \implies 348 = 6y \implies y = 58$

Step 3: Find the value of $x$

Substitute $y = 58$ back into the equation $x = 180 - y$: $x = 180 - 58 = 122$

Conclusion

The measures of the angles are:

• $m\angle 6 = 122^\circ$
• $m\angle 7 = 58^\circ$

Would you like more details or have any questions?

Additional Questions:

1. How do we verify that the calculated angles satisfy the original conditions?
2. What would happen if the angles were complementary instead of forming a linear pair?
3. Can we solve this problem using a different method, such as graphing the equations?
4. What is a linear pair, and why must their sum be 180 degrees?
5. How would the problem change if "twelve more" was replaced by "twelve less"?

Tip: When dealing with problems involving linear pairs, always remember their sum is 180 degrees!