Here is the solution to the exercise in the image:

We are tasked with matching the formulas for the measure of angle $\text{m}\angle 1$ in column A with the corresponding diagrams in column B.

Step-by-Step Matching

1. $\text{m}\angle 1 = a$:
   This formula applies to the case where $\angle 1$ is equal to the arc it intercepts directly (no division or subtraction).
   Match: Diagram (e)

2. $\text{m}\angle 1 = \frac{1}{2}(a - b)$:
   This formula is used when the angle is formed outside the circle by two intersecting secants or tangents.
   Match: Diagram (c)

3. $\text{m}\angle 1 = \frac{1}{2}a$:
   This formula applies when $\angle 1$ is an inscribed angle, and its measure is half the intercepted arc.
   Match: Diagram (a)

4. $\text{m}\angle 1 = \frac{1}{2}(b - a)$:
   This formula is used when the angle is formed outside the circle, with arcs $b$ and $a$ involved, and $b > a$.
   Match: Diagram (f)

5. $\text{m}\angle 1 = \frac{1}{2}b$:
   Similar to the case of $\frac{1}{2}a$, this formula is for an inscribed angle, but it intercepts arc $b$.
   Match: Diagram (d)

6. $\text{m}\angle 1 = \frac{1}{2}(a + b)$:
   This formula applies when the angle is formed inside the circle by two intersecting chords.
   Match: Diagram (b)

Final Answers

1. (e)
2. (c)
3. (a)
4. (f)
5. (d)
6. (b)

Would you like a detailed explanation for any specific match or a breakdown of the related geometry concepts?

Related Questions to Expand Understanding

1. What is the difference between an inscribed angle and a central angle?
2. How do you derive the formula for angles formed outside a circle?
3. Can $\text{m}\angle 1 = \frac{1}{2}(a + b)$ apply to angles outside the circle? Why or why not?
4. Why is the angle formed by intersecting chords inside the circle half the sum of intercepted arcs?
5. How do tangents and secants relate to the formulas for angles in circles?

Tip

Always identify whether the angle is inside, on, or outside the circle to determine the correct formula!