https://r2.erweima.ai/mathbot/image/1725864520081fkhgc6cq.jpg

In the image, we see two parallel lines with a triangle between them. The angles inside the triangle are marked as \( 6x \), \( 7x \), and \( 3x \). To solve for \( x \), we can use the following approach:

### Step-by-step Solution:

1. **Angle Sum in a Triangle**: The sum of the internal angles of any triangle is always \( 180^\circ \). In this case, the three angles inside the triangle are \( 6x \), \( 7x \), and \( 3x \). Thus, we can set up the equation:
   \[
   6x + 7x + 3x = 180^\circ
   \]

2. **Simplify the Equation**:
   \[
   16x = 180^\circ
   \]

3. **Solve for \( x \)**:
   \[
   x = \frac{180^\circ}{16} = 11.25^\circ
   \]

Thus, the value of \( x \) is \( 11.25^\circ \).

### Final Answer:
\[
x = 11.25^\circ
\]

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

Here are 5 related questions to expand on this topic:
1. What is the sum of the angles in a polygon with more than three sides?
2. How does the concept of alternate interior angles apply to parallel lines and triangles?
3. Can you find the value of each angle in the triangle after solving for \( x \)?
4. How would the solution change if the lines were not parallel?
5. How does the exterior angle theorem help in solving problems involving triangles?

**Tip**: In geometry, the sum of the interior angles of a triangle is always \( 180^\circ \), and this property can simplify many types of angle calculations.

Find the value of x in the triangle formed by parallel lines where the angles are given as 6x, 7x, and 3x.

Solve a geometry problem involving the sum of angles in a triangle. The triangle has angles expressed as 6x, 7x, and 3x, and uses the angle sum property to find x. Ideal for students in Grades 7-9.

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