Let's analyze the image to determine the measures of the angles $\angle 1$ through $\angle 6$.

Given angles and relationships:

• We know the following given angles:
  • $\angle 2 = 103^\circ$
  • $\angle 3 = 68^\circ$
  • $\angle 5 = 90^\circ$ (since it's marked as a right angle)
  • $\angle 6 = 41^\circ$ (given directly)

Step-by-step solution:

1. $\angle 1$: $\angle 1$ and $\angle 3$ are supplementary because they form a linear pair. Therefore: $\angle 1 = 180^\circ - 68^\circ = 112^\circ$

2. $\angle 4$: $\angle 4$ is an external angle for the triangle formed by $\angle 5$, $\angle 6$, and itself. The exterior angle theorem tells us that $\angle 4$ is equal to the sum of the opposite interior angles. So: $\angle 4 = \angle 5 + \angle 6 = 90^\circ + 41^\circ = 131^\circ$

3. $\angle 2$: Given in the diagram as $103^\circ$.

4. $\angle 3$: Given in the diagram as $68^\circ$.

5. $\angle 5$: Given as a right angle, so: $\angle 5 = 90^\circ$

6. $\angle 6$: Given as $41^\circ$.

Final values:

• $\angle 1 = 112^\circ$
• $\angle 2 = 103^\circ$
• $\angle 3 = 68^\circ$
• $\angle 4 = 131^\circ$
• $\angle 5 = 90^\circ$
• $\angle 6 = 41^\circ$

Let me know if you'd like further details or explanations. Here are some related questions:

1. What is the exterior angle theorem, and how is it applied?
2. How do you find supplementary angles in different geometric figures?
3. What other properties of triangles can help in finding missing angles?
4. How do linear pairs of angles relate to supplementary angles?
5. Can the sum of angles in any polygon be calculated? How?

Tip: In any triangle, the sum of the interior angles is always $180^\circ$. This can help when finding missing angles.