Let's analyze the image and solve the problems given.

Understanding the Diagram

The diagram shows two intersecting lines forming multiple angles. The angles are labeled with numbers (2, 5, 6, 8, etc.). The problem provides different conditions to find the measures of certain angles.

Solution for Part (a)

• Given: $\angle 6 = 58^\circ$
• To Find: $\angle 2, \angle 5, \angle 8$

Step-by-Step Solution for (a)

1. Finding $\angle 5$:
   $\angle 5$ and $\angle 6$ are vertically opposite angles. Vertically opposite angles are equal, so:
   $\angle 5 = \angle 6 = 58^\circ$

2. Finding $\angle 8$:
   $\angle 6$ and $\angle 8$ are supplementary angles because they are on a straight line (linear pair). Thus:
   $\angle 6 + \angle 8 = 180^\circ$ Substituting $\angle 6 = 58^\circ$, we get:
   $58^\circ + \angle 8 = 180^\circ$ $\angle 8 = 180^\circ - 58^\circ = 122^\circ$

3. Finding $\angle 2$:
   $\angle 2$ and $\angle 8$ are vertically opposite angles, so they are equal:
   $\angle 2 = \angle 8 = 122^\circ$

Solution for Part (b)

• Given: $\angle 8 = 144^\circ$
• To Find: $\angle 2, \angle 5, \angle 6$

Step-by-Step Solution for (b)

1. Finding $\angle 2$:
   $\angle 2$ and $\angle 8$ are vertically opposite angles. Thus:
   $\angle 2 = \angle 8 = 144^\circ$

2. Finding $\angle 6$:
   $\angle 6$ and $\angle 8$ are supplementary angles (linear pair). Thus:
   $\angle 6 + \angle 8 = 180^\circ$ Substituting $\angle 8 = 144^\circ$, we get:
   $\angle 6 = 180^\circ - 144^\circ = 36^\circ$

3. Finding $\angle 5$:
   $\angle 5$ and $\angle 6$ are vertically opposite angles, so:
   $\angle 5 = \angle 6 = 36^\circ$

Final Answers:

• Part (a):

  • $\angle 2 = 122^\circ$
  • $\angle 5 = 58^\circ$
  • $\angle 8 = 122^\circ$

• Part (b):

  • $\angle 2 = 144^\circ$
  • $\angle 5 = 36^\circ$
  • $\angle 6 = 36^\circ$

Would you like more details on any step?
Here are some related questions:

1. What are vertically opposite angles and why are they always equal?
2. How do supplementary angles relate to the concept of linear pairs?
3. What is the difference between adjacent and opposite angles in intersecting lines?
4. How can we find other angles in the diagram using similar properties?
5. Why is understanding angle relationships important in geometry?

Tip: Remember that vertical angles are always equal, and linear pairs are always supplementary!