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 $(n-2) \times 180^\circ$.