Let's analyze and solve each of these problems.

Problem 1:

Given:

• $m \angle ADC = 160^\circ$
• $m \angle EFB = 80^\circ$

We need to find the measure of $m \angle ABC$.

Since $\angle ABC$ is formed by the tangent and the chord $AB$, we can use the formula: $m \angle ABC = \frac{1}{2} \times m \angle EFB$

Substitute $m \angle EFB = 80^\circ$: $m \angle ABC = \frac{1}{2} \times 80^\circ = 40^\circ$

Answer for Problem 1: $m \angle ABC = 40^\circ$

Problem 2:

Given:

• $m \angle MKL = 220^\circ$
• $m \angle MNL = 140^\circ$

We need to find $m \angle MQL$.

Since $\angle MQL$ is formed by a tangent and a secant, we use the formula: $m \angle MQL = \frac{1}{2} (m \angle MKL - m \angle MNL)$

Substitute the values: $m \angle MQL = \frac{1}{2} (220^\circ - 140^\circ) = \frac{1}{2} \times 80^\circ = 40^\circ$

Answer for Problem 2: $m \angle MQL = 40^\circ$

Would you like a further breakdown of these steps or have any questions about the methods used?

Here are some related questions you might find helpful:

1. How do you find the angle between a tangent and a chord in a circle?
2. What is the relationship between a secant and a tangent when measuring angles?
3. How can you generalize the formula for any angle formed by a tangent and a chord?
4. What are the properties of angles formed by two intersecting chords?
5. How do different circle theorems apply to various configurations of chords, tangents, and secants?

Tip: Remember that angles formed by tangents and secants often use the average or half of arc measures in their formulas.