https://file.aiquickdraw.com/mathbot/file/17253676895614fz81fp5.jpg

https://file.aiquickdraw.com/mathbot/file/1725367689561obgszjv9.jpg

Let's analyze each figure one by one to find the value of \( x \).

### First Image

1. **Given Angles**:
   - Angle 1: \( 120^\circ \)
   - Angle 2: \( 50^\circ \)
   - Angle 3: \( 80^\circ \)
   - We need to find \( x \).

2. **Explanation**:
   - The three given angles (50°, 80°, and 120°) and \( x \) together form a quadrilateral.
   - The sum of the interior angles of any quadrilateral is \( 360^\circ \).

3. **Calculation**:
   \[
   120^\circ + 50^\circ + 80^\circ + x = 360^\circ
   \]
   \[
   250^\circ + x = 360^\circ
   \]
   \[
   x = 360^\circ - 250^\circ = 110^\circ
   \]

### Second Image

1. **Given Angles**:
   - \( \angle DAB = y \)
   - \( \angle ABC = 100^\circ \)
   - \( \angle CDA = 120^\circ \)
   - We need to find \( x \) and \( y \).

2. **Explanation**:
   - \( \angle DAB \) and \( \angle ABC \) are consecutive interior angles on the same side of a transversal cutting through parallel lines. Therefore, they are supplementary:
     \[
     y + 100^\circ = 180^\circ
     \]
     \[
     y = 180^\circ - 100^\circ = 80^\circ
     \]

   - Similarly, \( \angle DAB \) and \( \angle CDA \) are consecutive interior angles on the other side, and so:
     \[
     120^\circ + x = 180^\circ
     \]
     \[
     x = 180^\circ - 120^\circ = 60^\circ
     \]

### Summary
- In the **first figure**, \( x = 110^\circ \).
- In the **second figure**, \( x = 60^\circ \) and \( y = 80^\circ \).

Would you like more details or have any questions? Here are some related questions you might consider:

1. How can you determine the sum of interior angles in different polygons?
2. What are the properties of supplementary angles in a quadrilateral?
3. How do parallel lines and transversals influence angle calculations?
4. How can knowing one angle in a trapezoid help you find the others?
5. What other methods can be used to verify the results in angle-related problems?

**Tip**: Always remember that the sum of the interior angles in a polygon with \( n \) sides is \( (n-2) \times 180^\circ \).

Learn how to find the value of x in geometric figures involving angles and quadrilaterals. This problem explores the sum of interior angles in quadrilaterals and the properties of consecutive interior angles. Suitable for students in Grades 7-9.

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