Let's go through each of the six diagrams and determine the relationships between the angles $a$ and $b$, based on the options provided (complementary, supplementary, vertical, adjacent, alternate interior, corresponding, or alternate exterior).

1) First Diagram:

• The angles $a$ and $b$ are directly opposite each other, created by two intersecting lines. These angles are called vertical angles.
• Vertical angles are congruent, meaning $a = b$.

2) Second Diagram:

• The angles $a$ and $b$ are on opposite sides of a transversal, with both lines parallel. These angles are alternate interior angles.
• Alternate interior angles are congruent when the lines are parallel, so $a = b$.

3) Third Diagram:

• The angles $a$ and $b$ are on the same side of the transversal and in corresponding positions (one is above, one is below the lines).
• These are corresponding angles and are congruent when the lines are parallel, so $a = b$.

4) Fourth Diagram:

• The angles $a$ and $b$ are on opposite sides of the transversal but exterior to the two parallel lines.
• These are alternate exterior angles, and they are congruent, so $a = b$.

5) Fifth Diagram:

• The angles $a$ and $b$ are adjacent to each other, sharing a common arm.
• These are adjacent angles, and their sum is likely $180^\circ$, making them supplementary.

6) Sixth Diagram:

• The angles $a$ and $b$ are on opposite sides of a transversal, but one is on the inside (interior) and the other on the outside (exterior).
• These are alternate exterior angles, and they are congruent, so $a = b$.

Would you like more details or explanations on any of these concepts? Here are some follow-up questions that might deepen your understanding:

1. What is the difference between supplementary and complementary angles?
2. Why are vertical angles always congruent?
3. How can you prove that alternate interior angles are congruent using parallel lines?
4. Can adjacent angles ever be congruent? In what case?
5. How are corresponding angles used in real-world applications, like in construction?

Tip: Always remember that vertical angles are congruent and that alternate interior or exterior angles are congruent when lines are parallel.