Let's go step by step in solving the problems based on the given diagram and the angles provided:

Problem a:

• We are given that $\angle 6 = 58^\circ$.
• We need to find $\angle 2$, $\angle 5$, and $\angle 8$.

Step 1: Understanding the relationships

In this diagram, we have two intersecting lines, which means the following angle relationships apply:

1. Vertically opposite angles are equal.
2. Adjacent angles on a straight line are supplementary, meaning they sum to $180^\circ$.

Step 2: Finding $\angle 2$

• $\angle 2$ is vertically opposite $\angle 6$, so: $\angle 2 = \angle 6 = 58^\circ$

Step 3: Finding $\angle 5$

• $\angle 5$ is supplementary to $\angle 6$ since they are adjacent angles on the same straight line: $\angle 5 + \angle 6 = 180^\circ$ Substituting $\angle 6 = 58^\circ$: $\angle 5 = 180^\circ - 58^\circ = 122^\circ$

Step 4: Finding $\angle 8$

• $\angle 8$ is vertically opposite $\angle 5$, so: $\angle 8 = \angle 5 = 122^\circ$

Solution for a:

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

Problem b:

• We are given that $\angle 8 = 144^\circ$.
• We need to find $\angle 2$, $\angle 5$, and $\angle 6$.

Step 1: Finding $\angle 5$

• $\angle 5$ is vertically opposite $\angle 8$, so: $\angle 5 = \angle 8 = 144^\circ$

Step 2: Finding $\angle 6$

• $\angle 6$ is supplementary to $\angle 5$ since they are adjacent angles: $\angle 6 + \angle 5 = 180^\circ$ Substituting $\angle 5 = 144^\circ$: $\angle 6 = 180^\circ - 144^\circ = 36^\circ$

Step 3: Finding $\angle 2$

• $\angle 2$ is vertically opposite $\angle 6$, so: $\angle 2 = \angle 6 = 36^\circ$

Solution for b:

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

Would you like further clarification on any part of this explanation?

Relative Questions:

1. What is the relationship between vertically opposite angles?
2. How do adjacent angles on a straight line relate to each other?
3. How do you calculate an angle if you are given its supplementary angle?
4. How do intersecting lines affect angle measurements?
5. Can supplementary angles ever both be acute?

Tip:

Whenever two lines intersect, always look for vertically opposite angles and use the fact that adjacent angles are supplementary to simplify your calculations!