https://file.aiquickdraw.com/mathbot/file/1726082589744l1gka5j9.jpg

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!

∠6 and ∠7 form a linear pair. Twice the measure of ∠6 is twelve more than four times the measure of ∠7. Find the measure of each angle.

Learn how to solve for the measures of angles ∠6 and ∠7, which form a linear pair. This step-by-step guide uses algebraic equations and angle relationships to find the values. Suitable for students in Grades 8-10.

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