Let's analyze the diagram and solve for the angles denoted by lowercase letters.

Key Observations:

1. The triangle at the base (with angles $a$, $b$, and the 60°) is a straight-line triangle, meaning its angles must sum up to 180°.
2. The sum of all angles around a point, such as at the intersection of angles marked 20°, 45°, and $b$, is 180° because of the straight-line property.

Step-by-step Solution:

1. Finding $b$: At the point where angles 45°, 20°, and $b$ meet, the sum must be 180°. $b + 45^\circ + 20^\circ = 180^\circ$ Solving for $b$: $b = 180^\circ - 45^\circ - 20^\circ = 115^\circ$

2. Finding $a$: In the triangle where angles $a$, $b$, and 60° are located, the sum of the angles must be 180°. Now that we know $b = 115^\circ$, we can solve for $a$: $a + b + 60^\circ = 180^\circ$ Substituting the value of $b$: $a + 115^\circ + 60^\circ = 180^\circ$ Solving for $a$: $a = 180^\circ - 115^\circ - 60^\circ = 5^\circ$

3. Finding $C$: Since $C$ is on a straight line with angle $b$, their sum must also be 180°: $C + b = 180^\circ$ Substituting the value of $b$: $C + 115^\circ = 180^\circ$ Solving for $C$: $C = 180^\circ - 115^\circ = 65^\circ$

Final Results:

• $a = 5^\circ$
• $b = 115^\circ$
• $C = 65^\circ$

Would you like further clarification or more details on these calculations?

Here are some related questions to think about:

1. How do we apply the triangle sum property to different types of triangles?
2. What are the properties of supplementary angles in geometric figures?
3. How would you solve if an obtuse angle were involved in one of the triangles?
4. Can you explain the exterior angle theorem using this problem?
5. How would this problem change if one of the angles was unknown?

Tip: Always remember that the sum of angles in any triangle is always 180°. This is a fundamental rule in geometry!