Let's analyze the proof presented in the image you uploaded.

The goal is to fill in the gaps in the proof. The expression "a + b = 180°" and "b + c = 180°" both use certain properties of angles. You need to choose the correct reasoning for each part.

Here's a breakdown of the options based on the image:

1. a + b = 180° because - The two angles a and b are on a straight line, which makes them supplementary. This is explained by the property that angles which make a straight line sum to 180°. Therefore, this gap should be filled with "angles which make a straight line".

2. b + c = 180° because - Similarly, the angles b and c are supplementary as they form a straight line, so again, the reasoning should be "angles which make a straight line".

After this, the proof goes on to show:

• a + b = b + c.
• Simplifying this results in a = c, showing that vertically opposite angles are equal.

Therefore, the reasoning in the final part of the proof proves that vertically opposite angles are equal.

To summarize:

1. The correct choices to fill the gaps are:
   • "angles which make a straight line" for both "a + b = 180°" and "b + c = 180°".
2. The overall proof shows that a = c, which proves that vertically opposite angles are equal.

Let me know if you'd like further explanations on any part of this proof or have any other questions! Here are some related questions for you:

1. What is the general property of vertically opposite angles?
2. Can you explain why the angles that form a straight line are always supplementary?
3. What happens if two angles are supplementary but not vertically opposite?
4. How would you apply this proof to prove other geometric theorems?
5. How can we extend this concept to three-dimensional geometry?

Tip: Practice solving proofs by breaking down each step and focusing on the angle relationships involved.