The image contains a geometry problem related to parallel lines and angles, and the statements filled in explain the reasoning behind the calculations. Here's a summary of the work and the mathematical steps:

Problem Recap:

1. Given:

   • $m \angle 5 = 36^\circ$
   • $m \angle 4 = 42^\circ$
   • Lines $l$ and $k$ are parallel.
   • Line $n$ is the transversal intersecting $l$ and $k$.

2. Goal: Complete the reasoning behind angle relationships and validate that $m \angle 1 + m \angle 2 + m \angle 3 = 180^\circ$, confirming the sum of angles in a triangle.

Steps Explanation:

1. Angle Addition Property: $m \angle 2 + m \angle 4 + m \angle 5 = 42^\circ + 36^\circ + 78^\circ = 156^\circ$.

2. Relationship Between $\angle 4$ and $\angle 3$: $\angle 4$ and $\angle 3$ are alternate interior angles because line $n$ crosses parallel lines $l$ and $k$. Since these angles are congruent: $m \angle 3 = m \angle 4 = 78^\circ$

3. Relationship Between $\angle 1$ and $\angle 5$: $\angle 1$ and $\angle 5$ are corresponding angles due to the parallel lines. Hence: $m \angle 1 = m \angle 5 = 36^\circ$

4. Sum of Angles in the Triangle: $m \angle 1 + m \angle 2 + m \angle 3 = 36^\circ + 42^\circ + 78^\circ = 180^\circ$, confirming the triangle's angle sum property.

5. Final Answer: The sum of the angles $\angle 1, \angle 2, \angle 3$ confirms the interior angle property of a triangle.

Would you like to:

1. See how to derive properties of alternate interior or corresponding angles?
2. Review the relationships between transversal and parallel lines?
3. Understand the sum of angles in polygons beyond triangles?
4. Explore more examples of angle properties in geometric configurations?
5. Learn proofs related to the sum of angles in a triangle?

Tip:

Remember, parallel line properties and transversals are essential for solving problems in coordinate geometry and reasoning proofs. Always identify congruent or supplementary angles first!