https://file.aiquickdraw.com/mathbot/file/17265125358593yjwayic.jpg

To solve for angles \( A \), \( B \), and \( F \), let's break down the information from the image and use geometric principles.

### Given Information:
- The measure of angle \( \angle CDF = 157^\circ \) (external angle).
- The measure of angle \( \angle FCD = 42^\circ \) (internal angle).
  
We can use the fact that the sum of angles on a straight line is \( 180^\circ \) and the angles in a triangle sum to \( 180^\circ \).

### Step-by-Step Analysis:

1. **Finding \( \angle A \):**
   - \( \angle A \) is adjacent to \( \angle FCD = 42^\circ \). Since these two angles form a straight line, we can find \( \angle A \) using:
     \[
     \angle A + 42^\circ = 180^\circ
     \]
     Therefore:
     \[
     \angle A = 180^\circ - 42^\circ = 138^\circ
     \]

2. **Finding \( \angle F \):**
   - \( \angle F \) is opposite the external angle \( \angle CDF = 157^\circ \). The external angle theorem states that the external angle of a triangle is equal to the sum of the two non-adjacent interior angles. Therefore:
     \[
     \angle F + \angle B = 157^\circ
     \]

3. **Finding \( \angle B \):**
   - Finally, since the sum of the interior angles of a triangle is \( 180^\circ \), we have:
     \[
     \angle A + \angle F + \angle B = 180^\circ
     \]
     Substituting \( \angle A = 138^\circ \):
     \[
     138^\circ + \angle F + \angle B = 180^\circ
     \]
     Simplifying:
     \[
     \angle F + \angle B = 42^\circ
     \]
   Now, from the external angle theorem, we already know that \( \angle F + \angle B = 157^\circ \). Therefore:
   \[
   \angle F = 42^\circ \quad \text{and} \quad \angle B = 15^\circ
   \]

### Final Answers:
- \( \angle A = 138^\circ \)
- \( \angle B = 15^\circ \)
- \( \angle F = 27^\circ \)

Would you like further details or clarifications?

Here are 5 related questions to deepen your understanding:
1. How can we use the exterior angle theorem in other types of polygons?
2. What is the sum of interior angles for any triangle?
3. How do we calculate the angles formed by intersecting lines?
4. Can we always calculate angles in any polygon using the given methods?
5. How do alternate interior angles help in parallel lines?

**Tip:** The sum of angles around a point is always \(360^\circ\), a useful property when dealing with intersecting lines.

Learn how to find the angles A, B, and F in a triangle using the exterior angle theorem and linear pairs. This problem explains how to use geometric principles such as the sum of angles in a triangle and straight line angle properties. Suitable for Grades 6-8.

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