https://r2.erweima.ai/mathbot/image/17260393932135fggtxea.jpg

Let's begin by solving the geometry problems from the image you provided. The image contains four problems asking for the calculation of angle \( x \), with the information that \( L_1 \parallel L_2 \), indicating that the lines are parallel.

### First Problem (Top Left):
We are asked to find angle \( x \) given the parallel lines and a transversal.

- \( \angle A = 50^\circ \)
- \( \angle B = 100^\circ \)

Using the fact that \( \angle A \) and the unknown angle \( x \) are consecutive interior angles along the parallel lines and transversal, they must sum to \( 180^\circ \). Therefore:

\[
x + 100^\circ = 180^\circ
\]
\[
x = 180^\circ - 100^\circ = 80^\circ
\]

So, the answer for \( x \) is \( 80^\circ \), which corresponds to option \( c) \).

### Second Problem (Top Right):
Similarly, the problem asks to calculate \( x \).

- \( \angle A = 70^\circ \)
- \( \angle B = 100^\circ \)

Here, using the same rule of consecutive interior angles:

\[
x + 70^\circ = 180^\circ
\]
\[
x = 180^\circ - 70^\circ = 110^\circ
\]

So, the answer for \( x \) is \( 110^\circ \), corresponding to option \( d) \).

### Third Problem (Bottom Left):
We are provided with the angles:

- \( \angle A = 60^\circ \)

Again, applying the consecutive interior angles rule:

\[
x + 60^\circ = 180^\circ
\]
\[
x = 180^\circ - 60^\circ = 120^\circ
\]

Thus, the answer is \( 120^\circ \), which is option \( c) \).

### Fourth Problem (Bottom Right):
Given that:

- \( \angle A = 120^\circ \)

Applying the consecutive interior angle theorem once more:

\[
x + 120^\circ = 180^\circ
\]
\[
x = 180^\circ - 120^\circ = 60^\circ
\]

Therefore, the correct answer is \( 60^\circ \), corresponding to option \( a) \).

---

Would you like further details or clarifications on any of the solutions?

Here are 5 questions that could expand your understanding:
1. What is the role of alternate interior angles in parallel line problems?
2. How can you identify consecutive interior angles in more complex diagrams?
3. Can you use the same principles of parallel lines with more than two lines?
4. How do you apply these angle relationships when dealing with quadrilaterals formed by parallel lines?
5. What happens to angle relationships when lines are perpendicular rather than parallel?

**Tip**: Always look for relationships between angles such as corresponding, alternate interior, and consecutive interior angles when dealing with parallel lines.

Calculate the value of angle x given parallel lines and transversals in a series of geometry problems.

Solve geometry problems involving parallel lines and transversals to calculate unknown angles. Apply the consecutive interior angles theorem to find x.

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