Find the missing angles using geometry rules: sum of angles on a straight line is 180°, around a point is 360°, and vertically opposite angles are equal.

Problem: Finding the Missing Angles

The task involves solving for missing angles using three key geometric principles:
1. The sum of angles on a straight line is 180°.
2. The sum of angles around a point is 360°.
3. Vertically opposite angles are equal.

Let's solve each part step by step.

---

Part 1: The sum of angles on a straight line is 180°

#### Given:
- The sum of angles on a straight line is 180°.
- Two angles are given: 155° and 75°.
- We need to find the missing angles a and b.

#### Solution:
1. For angle \( a \):
- The angle \( a \) is on a straight line with the given angle \( 155° \).
- Using the property that the sum of angles on a straight line is 180°:
\[
a + 155° = 180°
\]
- Solve for \( a \):
\[
a = 180° - 155° = 25°
\]

2. For angle \( b \):
- The angle \( b \) is on a straight line with the given angle \( 75° \).
- Using the same property:
\[
b + 75° = 180°
\]
- Solve for \( b \):
\[
b = 180° - 75° = 105°
\]

#### Final Answers for Part 1:
\[
a = 25°, \quad b = 105°
\]

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Part 2: The sum of angles around a point is 360°

#### Given:
- The sum of angles around a point is 360°.
- Two angles are given: 230° and 85°.
- We need to find the missing angles c and d.

#### Solution:
1. For angle \( c \):
- The angle \( c \) is part of the angles around a point, along with the given angle \( 230° \).
- Using the property that the sum of angles around a point is 360°:
\[
c + 230° = 360°
\]
- Solve for \( c \):
\[
c = 360° - 230° = 130°
\]

2. For angle \( d \):
- The angle \( d \) is part of the angles around a point, along with the given angle \( 85° \) and another angle of \( 125° \).
- Using the property that the sum of angles around a point is 360°:
\[
d + 85° + 125° = 360°
\]
- Simplify:
\[
d + 210° = 360°
\]
- Solve for \( d \):
\[
d = 360° - 210° = 150°
\]

#### Final Answers for Part 2:
\[
c = 130°, \quad d = 150°
\]

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Part 3: Vertically opposite angles are equal

#### Given:
- Vertically opposite angles are equal.
- Two angles are given: 50° and 45°.
- We need to find the missing angles e and f.

#### Solution:
1. For angle \( e \):
- The angle \( e \) is vertically opposite to the given angle \( 50° \).
- By the property of vertically opposite angles:
\[
e = 50°
\]

2. For angle \( f \):
- The angle \( f \) is vertically opposite to the given angle \( 45° \).
- By the property of vertically opposite angles:
\[
f = 45°
\]

#### Final Answers for Part 3:
\[
e = 50°, \quad f = 45°
\]

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Final Answer:

\[
\boxed{a = 25°, b = 105°, c = 130°, d = 150°, e = 50°, f = 45°}
\]