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To solve this problem, we need to determine the value of the variable \( x \) and find the measures of each angle in the triangle \( \triangle PQR \).

The angles are given as:
1. \( \angle P = (2x - 12)^\circ \)
2. \( \angle Q = (2x + 2)^\circ \)
3. \( \angle R = x^\circ \)

### Step 1: Use the Triangle Angle Sum Property
The sum of the angles in a triangle is always \( 180^\circ \). Therefore, we can set up the following equation:

\[
(2x - 12) + (2x + 2) + x = 180
\]

### Step 2: Simplify and Solve for \( x \)

1. Combine like terms:
   \[
   5x - 10 = 180
   \]

2. Add 10 to both sides:
   \[
   5x = 190
   \]

3. Divide by 5:
   \[
   x = 38
   \]

### Step 3: Substitute \( x = 38 \) into Each Angle Expression

1. \( \angle P = 2x - 12 = 2(38) - 12 = 76 - 12 = 64^\circ \)
2. \( \angle Q = 2x + 2 = 2(38) + 2 = 76 + 2 = 78^\circ \)
3. \( \angle R = x = 38^\circ \)

### Final Answer
- \( x = 38 \)
- \( \angle P = 64^\circ \)
- \( \angle Q = 78^\circ \)
- \( \angle R = 38^\circ \)

Would you like further explanation or have any questions about this solution?

### Follow-Up Questions
1. How do we know the triangle angle sum property is always 180 degrees?
2. What would happen if the sum of the angles was not 180 degrees in a triangle?
3. How would this method change if the triangle was not a simple one (like if it was a right or obtuse triangle)?
4. Can you calculate the side lengths if only the angles are given in a triangle?
5. How can we verify if the calculated angles form an actual triangle?

### Tip
In any triangle, always start by checking if the sum of angles equals 180 degrees to validate your calculations.

For the figure shown on the right, find the value of the variable and the measures of the angles.

Learn how to find the value of the variable x and calculate the angles in a triangle using the Triangle Angle Sum Property. This problem involves algebraic expressions and is suitable for students in Grades 8-10.

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