https://file.aiquickdraw.com/mathbot/file/1726021956702dmi59pfa.jpg

To solve the variables in this diagram, we will examine the angles provided, as well as use properties of parallel lines and the geometry of intersecting lines (e.g., vertically opposite angles, corresponding angles, supplementary angles).

Here’s how to approach solving the problem:

1. **Angle Properties of Parallel Lines:**
   - When two parallel lines are cut by a transversal, the corresponding angles are equal, and the alternate interior angles are also equal.
   - Vertically opposite angles (formed by the intersection of two lines) are equal.
   - Adjacent angles that form a straight line (linear pair) sum to \(180^\circ\).

### Analyzing the Diagram:

- **Angle 1 (given as 42°)**
- **Angle 2 (given as 110°)**
- **Angle 3 (given as 124°)**

#### Key Observations:

1. **At the intersection at point X**:
   - \( \angle 1 \) is \( 42^\circ \).
   - Vertically opposite angles are equal, so the angle opposite \( \angle 1 \) is also \( 42^\circ \).

2. **At the intersection of lines u and w with line a**:
   - One of the angles is \( 115^\circ \).
   - Since adjacent angles along a straight line sum to \(180^\circ\), the adjacent angle to \(115^\circ\) will be:
   \[
   180^\circ - 115^\circ = 65^\circ.
   \]
   So, the angle opposite \( \angle 115^\circ \) is \(65^\circ\).

3. **At the intersection where angle \(124^\circ\) is given**:
   - The adjacent angle is:
   \[
   180^\circ - 124^\circ = 56^\circ.
   \]

4. **At the intersection where angle \(135^\circ\) is given**:
   - The adjacent angle is:
   \[
   180^\circ - 135^\circ = 45^\circ.
   \]

With this approach, you can use the same principles for every other set of angles and intersections.

Would you like more detailed steps, or should I solve specific variables from the diagram?

solve for all the variables in the geometric diagram

Solve for all variables in the geometric diagram by applying properties of parallel lines, vertically opposite angles, and supplementary angles. This solution includes key geometry concepts and theorems suitable for students in Grades 8-10.

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