To solve this problem, we need to find the measure of the smallest angle in a triangle $ABC$ where:

1. Angle $A$ is three times the measure of angle $B$.
2. Angle $C$ is twice the measure of angle $B$.

Let's define:

• $B = x$,
• $A = 3x$,
• $C = 2x$.

Since the sum of the angles in any triangle is $180^\circ$, we can write: $A + B + C = 180^\circ$ Substitute the values for $A$, $B$, and $C$: $3x + x + 2x = 180^\circ$ $6x = 180^\circ$ $x = 30^\circ$

Now, we can calculate each angle:

• $B = x = 30^\circ$,
• $A = 3x = 3 \times 30 = 90^\circ$,
• $C = 2x = 2 \times 30 = 60^\circ$.

The smallest angle in the triangle is therefore $B = 30^\circ$.

Answer:

The measure found by the artist for the smallest angle is $30^\circ$.

Would you like further details or have any questions?

Here are 5 related questions to consider:

1. What would the triangle look like if angle $A$ was four times angle $B$ instead?
2. How does the triangle’s shape change if angle $C$ was equal to angle $B$?
3. What would happen if we added an exterior angle to angle $A$ in this triangle?
4. How can we determine if this triangle is a right triangle based on these angles?
5. What other types of triangles could we create by adjusting the relationships between $A$, $B$, and $C$?

Tip: Always check if the sum of your calculated angles equals $180^\circ$ to ensure your solution satisfies the triangle angle sum property.