Let's break down the problems and solve them one by one.

For problems 1 to 4: Identify the relationship between angles

1. Problem 1:

   • Relationship between angles a and b.
   • These two angles are on the same straight line, so they form a linear pair (sum of angles = 180°).

2. Problem 2:

   • Angles a and b are opposite each other at the intersection of two lines.
   • These angles are vertical angles, which are congruent (equal).

3. Problem 3:

   • Similar to problem 2, angles a and b are opposite each other.
   • Again, this is a pair of vertical angles, which are congruent.

4. Problem 4:

   • Angles a and b are opposite each other where two lines intersect.
   • This relationship forms vertical angles, which are congruent.

For problems 5 and 6: Solve for $x$

5. Problem 5:

   • Given angles are 52° and 2$x$°. These two angles are vertically opposite, so they must be equal.
   • Set them equal to each other: $2x = 52°$ Solving for $x$: $x = \frac{52°}{2} = 26°$

6. Problem 6:

   • We have a 90° angle (indicated by the square) and two other angles that sum to 90°.
   • $(x + 12)° + 59° = 90°$
   • Simplifying: $x + 12 + 59 = 90$ $x + 71 = 90$ Solving for $x$: $x = 90 - 71 = 19°$

Let me know if you want more details on any of the steps!

Here are 5 related questions you could explore:

1. What is the sum of angles on a straight line, and why?
2. How can you use vertical angles to solve problems with intersecting lines?
3. What are adjacent angles, and how do they relate to complementary and supplementary angles?
4. In what situations do two angles form a linear pair?
5. How can you calculate missing angles when given complementary angle pairs?

Tip: Always remember, vertical angles are congruent (equal), and angles forming a linear pair are supplementary (sum to 180°).