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Let's solve each of the triangles for the value of \( x \) in the given figures.

### Problem 20:
This is a triangle where we know two angles and need to find the third. The angles given are \( 98^\circ \) and \( 41^\circ \).

- **Solution**: Use the triangle sum property, which states that the sum of the angles in a triangle is always \( 180^\circ \).
  \[
  x = 180^\circ - 98^\circ - 41^\circ
  \]
  \[
  x = 41^\circ
  \]

### Problem 21:
This is a straight line with angles given as \( 42^\circ \) and \( 79^\circ \). \( x \) is the angle adjacent to \( 79^\circ \).

- **Solution**: The angles on a straight line sum to \( 180^\circ \). Hence,
  \[
  x = 180^\circ - 79^\circ
  \]
  \[
  x = 101^\circ
  \]

### Problem 22:
This is a triangle with given angles \( 119^\circ \) and \( 46^\circ \). We need to find the third angle \( x \).

- **Solution**: Again, using the triangle sum property:
  \[
  x = 180^\circ - 119^\circ - 46^\circ
  \]
  \[
  x = 15^\circ
  \]

### Problem 23:
This triangle has angles \( 74^\circ \), \( 23^\circ \), and \( x \).

- **Solution**: Use the triangle sum property:
  \[
  x = 180^\circ - 74^\circ - 23^\circ
  \]
  \[
  x = 83^\circ
  \]

### Final Values:
- **Problem 20**: \( x = 41^\circ \)
- **Problem 21**: \( x = 101^\circ \)
- **Problem 22**: \( x = 15^\circ \)
- **Problem 23**: \( x = 83^\circ \)

Would you like any additional details or explanations on these problems?

### Further Questions:
1. How does the triangle sum property work in non-Euclidean geometry?
2. What is the relationship between exterior and interior angles of a triangle?
3. How can you determine if a triangle is acute, obtuse, or right using its angles?
4. What happens if the sum of angles in a polygon is more than \( 180^\circ \)?
5. Can the angles of a triangle be used to determine the type of triangle (equilateral, isosceles, scalene)?

### Tip:
When solving for angles in any polygon, always remember that the sum of the interior angles in a triangle is \( 180^\circ \), which can be a useful checkpoint in your calculations.

What is the value of x in each figure (triangles and angles)?

Learn how to find the value of x in different geometric figures using the triangle sum property and the sum of angles on a straight line. This solution covers problems where the given angles lead to calculations of missing angles. Suitable for students in Grades 6-8.

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