In this diagram, we have two intersecting lines, creating several pairs of angles. Let's solve for the missing angles based on the given values and the properties of vertical and supplementary angles.

Known Angles:

• $h = 123^\circ$
• $m = 81^\circ$

Steps:

1. Find $g$:

   • Angle $g$ is vertically opposite to $h$. Vertically opposite angles are equal, so: $g = h = 123^\circ$

2. Find $k$:

   • Angle $k$ is vertically opposite to $m$. Similarly, vertically opposite angles are equal, so: $k = m = 81^\circ$

3. Verify the relationships:

   • Angles $h$ and $m$ are supplementary because they are adjacent and form a straight line. The sum of supplementary angles is $180^\circ$: $h + m = 123^\circ + 81^\circ = 204^\circ$ This confirms that both angles are on a straight line, and the other pairs follow the same relationships.

Final Answer:

• $g = 123^\circ$
• $h = 123^\circ$
• $k = 81^\circ$
• $m = 81^\circ$

Would you like further details or explanations on any part of this solution?

Here are 5 related questions to deepen understanding:

1. What is the definition of vertically opposite angles?
2. How do you determine supplementary angles in a diagram?
3. Can two angles be supplementary if they are not adjacent?
4. How would the relationships change if the lines were not straight?
5. What other angle properties are involved in intersecting lines?

Tip: When solving for angles in a diagram, always check for vertical and supplementary angle pairs first!