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The image contains a set of geometric and analytical problems involving conditions of parallelism, perpendicularity, and the use of coordinates to solve for unknown values. Here’s a summary of the tasks:

1. **Problem 1**: Prove corollaries related to conditions of parallelism and perpendicularity using geometric and analytical elements.
   
2. **Problem 2**: Given the coordinates of the vertices of a parallelogram \((x, 4), (3, -1), (-2, -3), (-4, 2)\), find the value of the unknown variable \(x\).

3. **Problem 3**: Given the coordinates of the vertices of a parallelogram \((x, 5), (4, -2), (-3, -4), (-5, 3)\), find the value of the unknown variable \(x\).

4. **Problem 4**: Find the value of the unknown variable for the points \((-5, 4), (-1, 1), (y, 3)\) to be collinear.

5. **Problem 5**: Find the value of the unknown variable for the points \((-6, 5), (-2, 2), (4, y)\) to be collinear.

6. **Problem 6**: Two line segments, one passing through the points \((x, 6)\) and \((-6, -2)\), and the other through the points \((x, 5)\) and \((0, -2)\), are perpendicular. Find the value of the unknown variable \(x\).

7. **Problem 7**: Two line segments, one passing through the points \((x, 7)\) and \((-7, -3)\), and the other through the points \((x, 6)\) and \((1, -3)\), are perpendicular. Find the value of the unknown variable \(x\).

8. **Problem 8**: Prove that the points \(A(5, -2)\), \(B(-1, 4)\), and \(C(-5, 0)\) form the vertices of a right triangle.

If you'd like detailed steps to solve any of these problems, feel free to choose one, and I'll guide you through the process.

Do you want details or have any questions on this? 

### Five related questions:
1. How do you determine the slopes of two lines to check for perpendicularity?
2. What conditions must be met for three points to be collinear?
3. How do you apply the distance formula in coordinate geometry problems?
4. How can you prove that four points form a parallelogram using slopes or distances?
5. How can the concept of vectors help in solving geometric problems like these?

### Tip:
For checking collinearity of points, use the slope formula to ensure the slopes between consecutive points are equal.

Solve the set of problems related to parallelism, perpendicularity, and finding unknown variables based on geometric conditions in a parallelogram and collinear points.

Learn to solve problems involving finding unknown variables in a parallelogram, checking conditions for perpendicularity, and ensuring collinearity of points. Step-by-step guide for geometry students in Grades 9-11.

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