It seems that you've uploaded an image with some geometry questions involving parallel lines and angles. Let's examine the questions step by step.

In the image, you have four subproblems (a, b, c, and d) involving angles formed by two parallel lines cut by a transversal. Each problem seems to ask you to find the value of angle $x$.

I'll now solve each case, explaining the relevant geometric principles behind each solution:

a)

In this case, you are given an angle of $140^\circ$ and asked to find $x$. The angle $140^\circ$ is an exterior angle, and the two lines are parallel, so $x$ must be the alternate interior angle to the given one. The alternate interior angles are congruent, so: $x = 140^\circ.$

b)

Here, you are given an angle of $25^\circ$ and asked to find $x$. This is a case where consecutive interior angles are formed by the transversal. Consecutive interior angles are supplementary, meaning their sum is $180^\circ$. Thus: $x + 25^\circ = 180^\circ$ $x = 180^\circ - 25^\circ = 155^\circ.$

c)

In this case, you're given an angle of $115^\circ$ and asked to find $x$. The angle $115^\circ$ is an exterior angle, and since the lines are parallel, $x$ must be the corresponding angle. Corresponding angles are congruent, so: $x = 115^\circ.$

d)

In this case, you're given an angle of $117^\circ$ and asked to find $x$. Again, the lines are parallel, and $117^\circ$ is an exterior angle, which implies that $x$ is the alternate interior angle. Alternate interior angles are congruent, so: $x = 117^\circ.$

Would you like further clarification on any of these steps or concepts?

Here are some related questions for further exploration:

1. How do you determine whether two angles are alternate interior, corresponding, or consecutive interior angles?
2. Can you explain the relationship between alternate exterior angles in geometry?
3. What happens if the lines are not parallel? How would that affect the angle relationships?
4. How do you apply angle sum properties in polygons and triangles in similar geometric problems?
5. What are some practical applications of these geometric principles in real-life situations?

Tip: When dealing with parallel lines and transversals, always look for congruent angle pairs (alternate interior, corresponding) or supplementary angle pairs (consecutive interior).