https://r2.erweima.ai/mathbot/image/1726125962121b2keync5.jpg

To solve for \( \theta \) in the given parallelogram, let's analyze the information provided in the image.

### Key observations:
1. The figure is a parallelogram, which has the property that opposite angles are equal.
2. The angles on the adjacent sides of a parallelogram are supplementary (they sum to 180°).

### Given information:
- One angle is \( 2\theta + 115^\circ \).
- The adjacent angle is \( 30^\circ + 65^\circ = 95^\circ \).

### Steps to solve:
1. Since adjacent angles in a parallelogram are supplementary:
   \[
   (2\theta + 115^\circ) + 95^\circ = 180^\circ
   \]
2. Simplify the equation:
   \[
   2\theta + 115^\circ + 95^\circ = 180^\circ
   \]
   \[
   2\theta + 210^\circ = 180^\circ
   \]
3. Solve for \( \theta \):
   \[
   2\theta = 180^\circ - 210^\circ
   \]
   \[
   2\theta = -30^\circ
   \]
   \[
   \theta = -15^\circ
   \]

This suggests \( \theta = -15^\circ \), which may seem unconventional for an angle, but in context, it could represent a rotation or negative orientation.

Would you like further clarification or details? Here are some related questions:

1. What are the properties of a parallelogram that help solve for angles?
2. How do we know that adjacent angles are supplementary in this case?
3. Could the negative value for \( \theta \) imply a geometric interpretation?
4. How could we check the result by calculating other angles in the parallelogram?
5. How do properties of symmetry in the parallelogram influence the solution?

**Tip:** Always remember that the sum of interior angles in any quadrilateral is 360°, which can be useful when working with complex shapes.

Calcula el valor de θ en el paralelogramo dado con ángulos 2θ + 115° y 30° + 65°.

In this problem, we calculate the value of θ in a parallelogram using its angle properties. The angles given are 2θ + 115° and 30° + 65°. Follow the step-by-step algebraic approach to solve for θ. Suitable for students in Grades 8-10.

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