Looking at the diagram and based on the angles provided, I will help you determine the measures of the angles labeled from $m \angle 1$ to $m \angle 7$.

Step-by-step breakdown:

1. $m \angle 1$: $m \angle 1$ is supplementary to $80^\circ$ because they form a linear pair. Hence, $m \angle 1 = 180^\circ - 80^\circ = 100^\circ$

2. $m \angle 2$: $m \angle 2$ and $115^\circ$ are vertically opposite angles, so: $m \angle 2 = 115^\circ$

3. $m \angle 3$: $m \angle 3$ forms a linear pair with $m \angle 2$, so: $m \angle 3 = 180^\circ - 115^\circ = 65^\circ$

4. $m \angle 4$: $m \angle 4$ is vertically opposite to $65^\circ$ (from $m \angle 3$), so: $m \angle 4 = 65^\circ$

5. $m \angle 5$: This is a supplementary angle to $115^\circ$ and shares a straight line, so: $m \angle 5 = 180^\circ - 115^\circ = 65^\circ$

6. $m \angle 6$: $m \angle 6$ is part of a triangle. The sum of angles in a triangle is $180^\circ$. In the triangle with $61^\circ$, $90^\circ$ (right angle), and $m \angle 6$, we have: $m \angle 6 = 180^\circ - 61^\circ - 90^\circ = 29^\circ$

7. $m \angle 7$: $m \angle 7$ is a right angle, so: $m \angle 7 = 90^\circ$

Final Angle Measures:

• $m \angle 1 = 100^\circ$
• $m \angle 2 = 115^\circ$
• $m \angle 3 = 65^\circ$
• $m \angle 4 = 65^\circ$
• $m \angle 5 = 65^\circ$
• $m \angle 6 = 29^\circ$
• $m \angle 7 = 90^\circ$

Would you like further details or explanations on any specific part?

Related Questions:

1. How are vertically opposite angles formed and calculated?
2. Why do linear pairs of angles always add up to $180^\circ$?
3. Can you explain why the angles of a triangle always sum to $180^\circ$?
4. How do you calculate unknown angles in a right triangle?
5. How can angle relationships help solve complex geometric problems?

Tip: When dealing with a triangle, always remember that the sum of all interior angles is $180^\circ$.