The diagram shows intersecting lines forming four angles. We are given that $m \angle 2 = 109^\circ$ and need to find $m \angle 1$, $m \angle 3$, and $m \angle 4$.

Here’s how to solve it:

1. Vertical angles: $\angle 1$ and $\angle 2$ are vertical angles, meaning they are congruent. Therefore, $m \angle 1 = m \angle 2 = 109^\circ.$

2. Adjacent angles: $\angle 2$ and $\angle 3$ are adjacent and form a straight line (supplementary). The sum of supplementary angles is $180^\circ$. Hence, $m \angle 2 + m \angle 3 = 180^\circ.$ Solving for $m \angle 3$: $m \angle 3 = 180^\circ - 109^\circ = 71^\circ.$

3. Vertical angles: $\angle 3$ and $\angle 4$ are vertical angles, so they are congruent. Therefore, $m \angle 4 = m \angle 3 = 71^\circ.$

Final answers:

• $m \angle 1 = 109^\circ$
• $m \angle 3 = 71^\circ$
• $m \angle 4 = 71^\circ$

Would you like more details or have any questions?

Here are 5 related questions to expand your understanding:

1. What are vertical angles and why are they congruent?
2. What are supplementary angles and how do they relate to straight lines?
3. Can angles at an intersection ever be non-supplementary?
4. How would the problem change if $m \angle 2$ were different?
5. What happens if all the angles were equal? How would that affect the geometry?

Tip: Always remember, when two lines intersect, vertical angles are always equal, and adjacent angles are supplementary.