From the given image, we can see that two parallel lines $p$ and $q$ are intersected by a transversal. The angles are marked around the points of intersection. Specifically:

• Angle $g = 105^\circ$ is given.
• The task is to determine the other unknown angles $a, b, c, d, e,$ and $f$.

Step-by-step Solution:

1. Using the property of vertically opposite angles:

   • At the intersection of the transversal and line $q$, $g = 105^\circ$ implies $e = 105^\circ$ (vertically opposite angles).

2. Using the property of corresponding angles:

   • Since $p \parallel q$ and the transversal intersects them:
     • $g = b$ (corresponding angles), so $b = 105^\circ$.
     • $e = c$ (corresponding angles), so $c = 105^\circ$.

3. Using the property of supplementary angles:

   • On a straight line, angles on the same side of the transversal are supplementary:
     • $g + f = 180^\circ$, so $f = 180^\circ - 105^\circ = 75^\circ$.
     • $e + d = 180^\circ$, so $d = 180^\circ - 105^\circ = 75^\circ$.

4. Using the property of alternate interior angles:

   • Since $p \parallel q$:
     • $f = a$ (alternate interior angles), so $a = 75^\circ$.

Final Values:

• $a = 75^\circ$
• $b = 105^\circ$
• $c = 105^\circ$
• $d = 75^\circ$
• $e = 105^\circ$
• $f = 75^\circ$

Would you like a detailed explanation of any specific step?

Related Questions:

1. What are the properties of corresponding and alternate interior angles?
2. How can you verify that the lines are parallel using angle relationships?
3. What are supplementary angles, and how are they used in parallel line problems?
4. Can this method be applied to non-parallel lines? Why or why not?
5. How does the concept of transversals help in real-life geometric applications?

Tip:

Always check the relationships between angles (corresponding, alternate, or supplementary) when working with parallel lines and transversals.