To solve for the missing angles $g$, $h$, $k$, and $m$, we will apply the concepts of vertically opposite angles and supplementary angles.

Step-by-Step Solution:

1. Vertically Opposite Angles:

   • Angle $h$ and the given $127^\circ$ are vertically opposite angles. Therefore: $h = 127^\circ$

2. Supplementary Angles:

   • $g$ is adjacent to $127^\circ$ and forms a straight line with it. Since adjacent angles on a straight line sum to $180^\circ$, we can find $g$ as: $g = 180^\circ - 127^\circ = 53^\circ$

3. Vertically Opposite Angles (Again):

   • Angle $m$ and the given $98^\circ$ are vertically opposite angles. Therefore: $m = 98^\circ$

4. Supplementary Angles (Again):

   • Angle $k$ is adjacent to $98^\circ$ and forms a straight line with it. So, we can find $k$ as: $k = 180^\circ - 98^\circ = 82^\circ$

Final Answer:

$g = 53^\circ, \quad h = 127^\circ, \quad k = 82^\circ, \quad m = 98^\circ$

Would you like further details on any of these steps or explanations?

Here are five related questions to deepen your understanding:

1. How do vertically opposite angles relate to each other?
2. What is the sum of adjacent angles on a straight line?
3. If one angle of a straight line is $x$, how would you find the other angle?
4. Can you explain why supplementary angles sum to $180^\circ$?
5. How would the answer change if the given angles were different, say $120^\circ$ and $100^\circ$?

Tip: Always remember that vertically opposite angles are equal, and angles on a straight line always sum up to $180^\circ$.