In the figure, you're given the following information:

• $\angle 4 = 90^\circ$
• $\angle 5 = 50^\circ$
• $\angle 6 = 40^\circ$

The task is to find the measures of $\angle 1$, $\angle 2$, and $\angle 3$.

Step-by-step solution:

1. Use vertical angles:

   • $\angle 1$ and $\angle 5$ are vertical angles, so they must have the same measure: $\angle 1 = \angle 5 = 50^\circ$

2. Use linear pairs:

   • $\angle 4$ and $\angle 2$ form a linear pair, which means they add up to $180^\circ$: $\angle 4 + \angle 2 = 180^\circ$ Since $\angle 4 = 90^\circ$, we can solve for $\angle 2$: $90^\circ + \angle 2 = 180^\circ$ $\angle 2 = 180^\circ - 90^\circ = 90^\circ$

3. Use another linear pair:

   • $\angle 3$ and $\angle 6$ form a linear pair, so: $\angle 3 + \angle 6 = 180^\circ$ Given $\angle 6 = 40^\circ$, we can solve for $\angle 3$: $\angle 3 = 180^\circ - 40^\circ = 140^\circ$

Final answers:

• $\angle 1 = 50^\circ$
• $\angle 2 = 90^\circ$
• $\angle 3 = 140^\circ$

Would you like further details on any of these steps? Here are 5 related questions to explore further:

1. How do vertical angles help solve problems involving intersecting lines?
2. What is the linear pair theorem and how is it applied here?
3. Why do angles in a linear pair always add up to $180^\circ$?
4. How can you identify other angle relationships in this diagram?
5. What are some examples where vertical angles and linear pairs are used in real-world applications?

Tip: In any geometry problem involving intersecting lines, always look for vertical angles and linear pairs—they can help simplify the solution quickly!