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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.

Um artista fará uma pintura de uma figura geométrica triangular, a qual será chamada de ABC. O ângulo A terá a medida equivalente ao triplo do ângulo B, e o ângulo C será o dobro do ângulo B. Para iniciar sua arte, o artista precisa determinar a medida do menor ângulo desse triângulo.

Discover how to find the smallest angle in a triangle ABC where angle A is three times angle B, and angle C is twice angle B. This problem covers essential geometric concepts and the triangle angle sum theorem, ideal for students in Grades 7-9.

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