Find the lettered angles in each of the following.

The angle $340^\circ$ is a reflex angle. The interior angle at that vertex is the difference between $360^\circ$ and the reflex angle.

Draw an auxiliary line through the vertex where angles $f$ and $80^\circ$ are located, parallel to the two main parallel lines. This line divides the angle at that vertex into two parts.
The "M-rule" states that the sum of angles pointing in one direction (e.g., left) equals the sum of angles pointing in the opposite direction (e.g., right).
In this figure, the angles $e$ and $f$ point to the left, and the angles $80^\circ$ and $20^\circ$ (the interior angle at the bottom) point to the right.
Therefore, we have:

This equation has two unknowns. Let's re-examine the diagram for a clearer interpretation of $f$ and $80^\circ$.

Upon closer inspection, the angle $80^\circ$ is shown adjacent to $f$ at the same vertex, and both are inside the "V" shape. This is an unusual way to label. The most common interpretation for such a diagram is that the angle formed by the two transversals at the middle vertex is the sum of $f$ and $80^\circ$. So, the angle at the middle vertex is $(f+80^\circ)$.

Let's re-apply the "M-rule" with this interpretation:
Angles pointing left: $e$ and the interior angle at the bottom ($20^\circ$).
Angles pointing right: The angle at the middle vertex, which is $(f+80^\circ)$.

This still leaves two unknowns.

Let's consider another common interpretation for such diagrams: the $80^\circ$ is an angle between the top transversal and the middle transversal, and $f$ is an angle between the middle transversal and the bottom transversal. In this case, the $80^\circ$ and $f$ are separate angles at different "bends" of the zigzag. However, the diagram clearly shows $f$ and $80^\circ$ at the same vertex.

Given the way $f$ and $80^\circ$ are drawn at the same vertex, with $f$ having a circle around it (often indicating the angle itself) and $80^\circ$ also marked inside, it implies that the angle $f$ is $80^\circ$. This is the only way to get a unique solution for $e$.

If $f = 80^\circ$, then using the "M-rule" where angles pointing left ($e$ and $20^\circ$) equal angles pointing right ($f$):

Substitute $f = 80^\circ$:

Let's verify this with the auxiliary line method.
Draw an auxiliary line through the vertex of $f$ parallel to the main parallel lines. This line divides $f$ into two parts, $f_1$ (top part) and $f_2$ (bottom part).
$f_1$ is alternate interior to $e$. So $f_1 = e = 60^\circ$.
$f_2$ is alternate interior to the $20^\circ$ angle at the bottom. So $f_2 = 20^\circ$.
Then $f = f_1 + f_2 = 60^\circ + 20^\circ = 80^\circ$.
This is consistent with our assumption that $f=80^\circ$.

Therefore, for Fig. 9.19:

• $e = \mathbf{60^\circ}$
• $f = \mathbf{80^\circ}$

Figure 2 (Fig. 9.20)

The arrows on the lines indicate that they are parallel.

Draw an auxiliary line through the vertex where angles $g$ and $h$ are located, parallel to the two main parallel lines. This line divides the angle at that vertex into two parts.
The "M-rule" states that the sum of angles pointing in one direction (e.g., left) equals the sum of angles pointing in the opposite direction (e.g., right).
In this figure, the angles $58^\circ$ and $42^\circ$ point to the left.
The angle at the middle vertex, which is $(g+h)$, points to the right.
Therefore, we have:

The circle around $h$ indicates that $h$ is the angle inside the "V" shape at that vertex. The angle $g$ is shown adjacent to $h$, also inside the "V" shape. This implies that the total angle at that vertex is $g+h$.
However, the diagram also shows $g$ as a separate angle. In such diagrams, if two letters are at the same vertex, they usually represent parts of the total angle.
Let's assume $g$ and $h$ are the two parts of the angle at the middle vertex.
The angle $g$ is alternate interior to the $58^\circ$ angle.
The angle $h$ is alternate interior to the $42^\circ$ angle.

Draw an auxiliary line through the vertex of $g$ and $h$, parallel to the top and bottom parallel lines.
The angle $g$ and the angle $58^\circ$ are alternate interior angles.

The angle $h$ and the angle $42^\circ$ are alternate interior angles.

This matches the sum derived from the "M-rule" in Step 1.

Therefore, for Fig. 9.20:

• $g = \mathbf{58^\circ}$
• $h = \mathbf{42^\circ}$

Final Answers:
For Fig. 9.19:

• $e = \boxed{\mathbf{60^\circ}}$
• $f = \boxed{\mathbf{80^\circ}}$

For Fig. 9.20:

• $g = \boxed{\mathbf{58^\circ}}$
• $h = \boxed{\mathbf{42^\circ}}$