This image shows various geometric problems involving triangles and lines with specific angles marked. Let's analyze each one and see if we can solve for the unknown variables based on angle relationships and triangle properties.

Problem Analysis

1. Problem 3:

   • Two intersecting lines form an angle relationship.
   • Given angles: 140° and 157°, and the variable $x$.
   • We can use angle sum and linear pair properties to solve for $x$.

2. Problem 4:

   • This shows a transversal crossing two parallel lines, with an angle of 85° and the variables $x$ and $y$.
   • Alternate interior and corresponding angle theorems can help solve for $x$ and $y$.

3. Problem 5:

   • A triangle with an interior angle of 35° and the variables $x$ and $y$.
   • Using the triangle angle sum property, we can solve for the unknowns.

4. Problem 6:

   • Another triangle setup with two marked angles, 65° and 30°, and variables $x$ and $y$.
   • Similar to the previous problem, we can apply the triangle angle sum property.

If you'd like, I can go through each problem step-by-step and solve for each variable. Let me know if you're interested in that level of detail or if you have any specific questions!

Related Questions

1. How do you determine angles in a triangle using the angle sum property?
2. What is the alternate interior angle theorem, and how does it apply to parallel lines?
3. How do you use linear pairs to find missing angles?
4. What strategies are helpful for solving multi-variable angle problems in geometry?
5. How do you confirm if two lines are parallel based on angle relationships?

Tip: Remember that the sum of angles in any triangle is always 180°. This property is often helpful in solving for unknown angles in geometric problems.