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The problem involves 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 \( B = x \) (in degrees).

Then:
- \( A = 3x \)
- \( C = 2x \)

Since the sum of the angles in any triangle is \( 180^\circ \), we can write the equation:

\[
A + B + C = 180^\circ
\]

Substituting the values:

\[
3x + x + 2x = 180^\circ
\]

This simplifies to:

\[
6x = 180^\circ
\]

Dividing both sides by 6:

\[
x = 30^\circ
\]

Now, we can find the measures of all angles:
- \( B = x = 30^\circ \)
- \( A = 3x = 3 \times 30 = 90^\circ \)
- \( C = 2x = 2 \times 30 = 60^\circ \)

The smallest angle in the triangle is \( B = 30^\circ \).

**Answer:** The measure found by the artist for the smallest angle is \( 30^\circ \).

Would you like more details on the steps or have any questions?

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Here are some related questions to consider:

1. How does the sum of angles change in a non-Euclidean triangle?
2. What would happen if angle \( C \) was four times \( B \)?
3. Can this problem be solved using trigonometric identities?
4. What other relationships between angles can form a right triangle?
5. How can this approach be applied to polygons with more than three sides?

**Tip:** When working with angles in a triangle, remember that their sum is always \( 180^\circ \). This property is fundamental in geometry and helps in solving many angle-based problems.

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. A medida encontrada por esse artista será?

Discover how to find the smallest angle in triangle ABC, where angle A is three times angle B, and angle C is twice angle B. This solution covers geometric concepts like the Triangle Angle Sum Property. Suitable for students in Grades 7-9.

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